JC-NRLF 


ma 


REESE   LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 


Deceived 


'lass  No. 


THE  TRANSITION-CURVE 
FIELD-BOOK. 


CONTAINING   FULL   INSTRUCTIONS   FOR   ADJUSTING   AND 

LOCATING  A   CURVE  NEARLY  IDENTICAL  WITH  THE 

CUBIC  PARABOLA    IN    TRANSITION  BETWEEN 

ANY   CIRCULAR    RAILROAD    CURVE 

AND    TANGENT. 


SIMPLIFIED  IN   APPLICATION  BY  THE  AID  OF 

A    GENERAL    TABLE, 

AND    ILLUSTRATED    BY    RULES    AND    EXAMPLES    FOR 

VARIOUS    PROBLEMS    OF    LOCATION: 

ALSO    TABLES   OF  RADII,    SINES,   TANGENTS,  VERSINES 
AND  EXTERNAL  SECANTS. 


BY 

C.E. 


NEW  YOR  K.- 
JOHN   WILEY   &    SONS, 

53   EAST  TENTH   STREET. 
1891. 


COPYRIGHT,  1891, 

BY 
JOHN   WILEY   &   SONS. 


FEERIS  BROS., 
ROBERT  DRUMMOND,  printers, 

Electrotypes  82G  pearl  street, 

444  &  446  Pearl  Street,  °    „       Yorkt 

New  York. 


PREFATORY. 


THE  aim  of  this  Field-book  is  to  furnish  plain, 
practical  rules  and  examples  for  guidance  in  ad- 
justing and  locating  a  curve,  nearly  identical  with 
the  Cubic  Parabola,  as  a  Transition  Curve  in  con- 
necting circular  curves  with  tangents. 

In  the  investigation  of  the  principles  upon  which 
the  rules  are  based,  it  will  be  seen  that  with  data 
consisting  in  great  part  of  familiar  approximations 
used  in  circular-curve  location,  and  with  no  mathe- 
matics beyond  a  little  algebra  and  trigonometry, 
practically  exact  results  are  reached  in  regard  to 
laws  of  the  Transition  Curve  and  its  relations  to 
circular  curves. 

By  means  of  Transition  Curves  of  less  than  i$° 
of  central  angle,  tangents  can  be  connected  with  all 
circular  curves  used  on  railroads,  their  combined 
location  presenting  little,  if  any,  more  difficulty 
than  ordinary  circular-curve  location. 

The  chords  of  each  Transition  Curve  are  all  of 
the  same  length,  but  this  length  varies  with  each 
different  rate  of  curvature  of  the  connecting  circular 
curve.  By  the  use  of  the  General  Table,  however, 
which  applies  to  any  chord  length,  the  Transition 
Curve,  whether  laid  off  by  angles  or  ordinates,  is 
readily  adjusted  so  as  to  connect  with  any  circular 
curve,  the  two  curves  having  the  same  rate  of  cur- 
vature, and  a  common  tangent,  at  their  point  of 
connection. 

All  of  the  Transition  Curves  considered,  when 
laid  off  by  angles,  begin  with  a  2-minute  deflec- 


IV  PREFATORY. 

tion  angle  for  the  first  chord,  irrespective  of  the 
length  of  the  chord  or  the  degree  of  the  connect- 
ing circular  curve  ;  and,  consequently,  the  field- 
work  of  location  of  all  Transition  Curves  laid  off  by 
angles  and  with  the  same  number  of  chords  is  the 
same,  except  for  the  change  of  the  chord  length  in 
connecting  with  circular  curves  of  different  degrees 
of  curvature,  every  connecting  circular  curve  re- 
quiring a  special  chord  length  in  the  Transition 
Curve. 

The  ordinates  for  laying  off  any  Transition  Curve 
all  depend  upon  its  special  chord  length,  but  each 
ordinate  of  every  Transition  Curve  can  be  obtained 
from  a  single  line  of  the  table  by  simply  multiply- 
ing a  quantity  taken  from  it  by  the  special  chord 
length  in  decimals  of  100  feet. 

It  was  at  first  intended  to  give  three  tables  for 
Transition  Curves,  beginning  with  deflection  angles 
for  the  first  chord  of  i  minute,  2  minutes,  and  3 
minutes,  respectively  ;  but  the  former  and  latter 
were  laid  aside  for  the  reason  that  the  advantages 
of  increased  convenience  and  diminished  liability 
to  error,  in  having  a  single  table  to  refer  to,  en- 
tirely outweigh  that  of  being  able  to  make  rather 
nicer  adjustments  in  a  few  cases,  which  is  about  all 
that  would  be  gained  by  using  the  three  tables  in- 
stead of  one.  For  such  Transition  Curves  as  are 
needed  in  practice  the  table  given,  based  upon  the 
initial  deflection  of  2  minutes,  seerns  fo  be  all  that 
is  required. 

The  word  "  connect,"  as  applied  to  the  junction 
of  Transition  and  Circular  curves,  means  that  at 
their  point  of  junction  the  two  curves  have  always 
a  common  tangent,  and,  unless  otherwise  specified, 
the  same  rate  of  curvature. 


PREFATORY.  V 

The  expression  "central  angle,"  when  applied 
to  the  Transition  Curve,  means  the  total  curvature 
between  the  tangent  at  the  initial  point  and  that 
at  the  other  point  considered. 

Although  the  law  governing  the  relations  be- 
tween deflection  angles  from  the  initial  station  of 
Transition  Curves  has  been  suggested  heretofore, 
it  may  be  as  well  to  say  that  when  first  brought  to 
the  attention  of  the  writer  it  was  through  results 
then  already  reached  and  embodied  in  the  present 
General  Table, — and  also  that  he  alone  is  responsi- 
ble for  any  new  departures  either  in  the  matter  or 
the  manner  of  this  Field-book. 


THE   TRANSITION-CURVE 
FIELD-BOOK. 


GENERAL  PROPOSITION. 

If  a  tangent  SK  and  a  circular  arc  DA  are  to  be 
connected  by  a  cubic  parabola  SEL  of  central 
angle  a,  the  two  curves  to  have  a  common  tangent 
Z,T  and  the  same  rate  of  curvature  at  their  junction 
Z,  the  adjusted  gap  CD  between  tangent  and  circu- 
lar curve  will  be  one  third  of  the  middle  ordinate 
FD  of  the  circular  arC  of  201,  and  one  fourth  of  the 
terminal  ordinate  LK  of  the  parabola, 

i 


2  THE    TRANSITION-CURVE    FIELD-BOOK. 

Draw  the  circular  arcAV\  the  chord  VL  sub- 
tending the  central  angle  2a\  and  the  tangent  LT. 
The  cubic  parabola  of  central  angle  a  and  terminal 
tangent  LT  will  have  its  initial  tangent  SK  par- 
allel to  VL.  Through  F,  the  middle  of  the  chord 
LV,  draw  the  radius  OD  and  prolong  it  to  C,  mak- 

FD 

ing  CD  =  -  —  .     At  L  and  V  lay  off  LK  and  FS 


perpendicular  to  ZFand  equaj  to  ^C.  Draw  .S^T, 
which  will  be  parallel  to  L  V,  and  will  pass  through 
C.  In  a  cubic  parabola  SEL  the  ordinate  CYs  mid- 
way between  £  and  K  is  to  the  ordinate  LK  at  K 


and     CE=~. 
o 


~       is  to  (SJT)3,  or 


But  by  construction 


Hence  CZ>  =  zCE,  and,  cowsequently,  the  cubic 
parabola  SEL  will  pass  midway  between  C  and  J9. 

Denoting  the  middle  ordinate  FD  of  20:  by  m, 
the  gap  (7Z>  by  ^,  and  the  end  ordinate  Z^T  by  Jt:; 
by  substitution  in  last  equation, 


m       x 


which  was  to  be  proved. 

Also   if   another  circular  arc  connects  with  the 
same  parabola  under  the  same  conditions  at  any 


THE    TRANSITION-CURVE    FIELD-BOOK.  3 

other  point  Z',  the  same  relations  between  the  two 
curves  exist;  and  using  similar  notation, 

m'       x' 


Denoting  the  radius  of  the  circular  arc  LD  by 
,    the    parabola    length     SL    by   n,   and    taking 

=  —-y  from  similar  triangles,     ON 

n      n  t? 

Jt  :  —  ::  —  :  m,     and     m  =  r-=. 
4       2  O./E 


Substituting  for 


45840' 

whence 

_m_   Z>V 

=  3  ~~  ^37520'  ' 
jpo  =  i37520^  /  v 

fi (5) 

-M^f 

and 


34380- 

By  similar  deduction  for  the  circular  curve  con- 
necting with  the  parabola  at  Z', 


- 
whence 


4  THE    TRANSITION-CURVE    FIELD-BOOK. 

But  considering  x   and   xf  as   ordinates   to   the 
cubic  parabola,  approximately, 

x  :  x'  ::  n*  :  nn  ......     (7) 

Therefore 

D°n*  :D'°nn::n*:n'\ 
and 

DQ  :D'°  ::n:n'  ......     (8) 

The  central  angles  a  and  a'  turned  in   the  pa- 
rabola from  S  to  L  and  Z',  respectively,  are  meas- 

n          n* 
ured  by  the  circular  arcs  of  the  length  -  and  —  , 

and  of  degrees  of  curve  JD°  and  DfQ  per  100  feet. 
Therefore 

D°n 
a  —  --  ,     ......     (9) 

200 


200 

and 

a:  a'  ::  D°n:D'°n'. 

Multiplying  alternate  terms  of  (8)  by  n  and  «', 

D°n:£>'°n'::  n*'.n'\ 
Hence 


(10) 


If  N  signifies  the  number  of  chord  lengths  c  in 
^,  and  N'  the  number  of  the  same  chord  lengths 
c  in  ;/,  then 

n  —  cN    and     n'  —  cN'\ 
and  substituting  these  values  of  n  and  n'  in  (7), 


THE    TRANSITION-CURVE    FIELD-BOOK.  5 

(10),  and   (8),  respectively,  the  following  propor- 
tions result: 

x:    x'  ::  N*  :  N'\  ....     (n) 

a\   a'  ::N*:  Nn,  ....     (12) 

Z>°  :  D'°  ::  N  :  N' (13) 

Or,  in  words: 

I.  The  ordinates  at  the  chord  stations  are  as  the 
cubes  of  the  station  numbers. 

II.  The  central  angles  from  S  to  the  chord  sta- 
tions are  as  the  squares  of  the  station  numbers. 

III.  The   degrees  of   curves  connecting   at    the 
chord  stations  are  directly  as  the  station  numbers. 

(T 
In  Fig.  i,  —  is  the  sine  of  the  deflection  angle^ 

O  (JT 

at  6*  between  tangent  SK and  chord  SL.     Call  this 
deflection  angle  A. 

The  sine  of  one  half  the  central  angle  a  turned 

in  the  distance  —  of  circular  arc,  or  n  of  parabola, 
FD 

lszir 

But 

CG  =  23     and     FD  =  $q\ 
and  taking 


A  **  iW. 

sin  A  =  20  -±  -  —  --, 
2        n 


,  n       o<7 

sin  \a  =  -iff  -. —  —  — . 
2        n 


6  THE    TRANSITION-CURVE    FIELD-BOOK. 

Hence 

si*n  A  :  2  sin  \a  ::  4  :  12, 

or  approximately 

A\  a::  i  :  3, 

and 


In  words  :  The  deflection  angle  at  S  between 
tangent  Sl£  and  any  chord  station  is  one  third  of 
the  corresponding  central  angle. 

This  relation  between  the  central  and  deflection 
angles  here  roughly  deduced  by  approximations 
is,  when  applied  to  the  Transition  Curve  differing 
slightly  from  the  cubic  parabola,  almost  exact  up 
to  14°  24'  of  central  angle,  the  limit  of  the  General 
Table. 

The  greatest  angular  error  is  -^^  of  one  minute, 
or  9  seconds;  this  occurring  in  the  deflection  angle 
from  S  to  station  12,  which  calculated  by  the 

•yr 

formula    ^  =  tang  A    gives    4°  47'iBoJV>  instead  of 

4°  48'  as  recorded  in  the  table.  This  discrep- 
ancy affects  the  ordinate  at  station  12  of  a  Transi- 
tion Curve  500  feet  long  about  0.02;  and  for  one 
of  10  stations  500  feet  long  the  angular  error  re- 
duces to  2  seconds,  affecting  the  end  ordinate  less 
than  o.oi. 

The  quantities  in  lines  X  and  Y  were  originally 
obtained  approximately  from  the  formulas  X  = 

i \R  versin  a  and  Y=  -  -{-  R  sin  a,  after  assuming  c 
and  calculating  D°  by  transposing  (9)  and  a  from 


THE    TRANSITION-CURVE    FIELD-BOOK.  f 

(12)  on  the  basis  of  a  —  6  minutes.     Then  equations 
-     j^-/  x 

—  =  tang  i  and  —  =  tang  A   determined  the 


—  —  —  — 

angles  $  and  A  as  now  recorded  in  the  General 
Table.  Afterward  the  quantities  of  lines  X  and 
Y  were  recalculated  by  summing  the  products  of 
each  chord  into  the  sine  and  cosine,  respectively, 
of  its  angle  of  inclination  to  the  tangent  SK. 

The  ordinates  and  ordinate  distances  X  and  Y 
thus  obtained  are  for  the  chord  stations  of  a 
Transition  Curve  of  100  foot  chords;  but  for  any 
other  chord  length  the  corresponding  ordinates 
and  ordinate  distances  .rand  y  are  directly  propor- 

tional.    As  -  is  the  tangent  of  the  deflection  angle 

from  S  to  the  station  corresponding  to  x,  it  follows 
that,  when  the  same  chord  length  is  used,  each 
station  must  necessarily  fall  in  the  same  place 
whether  laid  off  by  ordinates  or  by  deflection 
angles. 

The  length  of  any  long  chord  (/)  is 
•   Y-  Y'         c 

/=  -      -rr-    X  --  ,    .      .      .      .      (15) 
COS  I  100 

in  which  Y  and  Y'  are  at  the  end  stations  of  /,  and 
/  is  the  angle  of  its  inclination  to  SK. 

EXAMPLE.   Long  chord  from  5  to  n  when  c  —  40. 

Taking  Fn,  F5,  and  /for  (5),  (11)  from  table: 

/-  I095.I3-499-9I  x   40  __  595-22 

cos  6°  42'"      <  ioo~  > 


If  /  begins  at  S  (see  Fig.  i),  formula  (15)  becomes 

Y  c 

/=  -    —  ;  X  —  . 
cos  A       100 


THE    TRANSITION-CURVE    FIELD-BOOK. 

EXAMPLE.   Long   chord    S  to    10    when    c  =  25. 
Taking  F10  and  J10  from  table, 


_       =         66. 

ioo 


The  length  of  tangent  /  from  S  to  its  intersection 
with  tangent  /'  from  L  is 

t=  (Y-  ^cotang  a)  —  ,          .     (16) 
7 


ioo 


in  which  XY  and  a  refer  to  the  station  at  L. 
EXAMPLE.  L  at  station  10;  c  =  25. 
Taking  F10,  AT10,  and  alo  from  table, 

/  =  99^97  ~  5^.053  X  5-671281  = 
4 

The  length  of  tangent  /'  from  L  to  its  intersec- 
tion with  tangent  SK  is 


^  c  i     ^ 

=  -• —  x  — ,    .    .    .    .    (17) 

sin  a       ioo 


in  which  X  and  a  refer  to  the  station  at  Z. 
EXAMPLE.  L  at  station  10;  c  —  25. 
Taking  XIQ  and  a10  from  table, 

t,  _.    58-053     y    25   _ 

-x- 


In  the  triangle  of  sides  /,  /,  and  f  (STL,  Fig.  i), 
the  angles  opposite  these  sides  are  respectively 
180  —  a,  zA,  and  A,  and  the  correctness  of  the  re- 

c\ 

lation  A  =  —  may    be    tested   by    substituting   for 
o 


THE   TRANSITION-CURVE    FIELD-BOOK.  9 

zA  arid  A  their  values  —  and  —  ,  and  recalculating 

o  o 

t  and  /',  with  /and  the  angles  as  data,  by  the  fol- 
lowing trigonometrical  proportions: 


2 

sin  (180°  —  a)  :  sin  —a  ::/:/; 
o 

sin  (180°  -  a)  :  sin  -  ::  /:  /'. 
3 

Whence,  taking  alQ  from  table, 

/sin  6°  40'    _  249.66  X  .1160929 

t  irr          ;  Q—    —  -     —  zi^    I  O  O  .  O  I  , 

sin  10  .1736482 

and 

_  /sin  3°  20'  _  249.66  X  .0581448  _ 
"ihiTo0""  ~~~~ 


giving  practically  the  same  values  as  before. 

As  A  =  —  ,  from  proportion  (10), 
«5 

AN-  AN'::N*:N'\ 
making 

N'  =  i; 


and  with  the  initial  deflection  angle  ^  =  2  minutes, 


Making  JV  =  i,  2,  3,  etc.,  the  angles  of  line  zf  of 
the  table  result;  and  multiplying  these  by  3  gives 
the  corresponding  angles  of  line  a. 


10  THE    TRANSITION-CURVE    FIELD-BOOK. 

The    remaining    angles    of  inclination    /  of    the 
chords  may  be  obtained  by  the  formula 

J^  J£7' 

y  _  y,  =  tang  /,  (19) 

in  which  X9  and  Y9  are  from  the  column  giving 
the  station  at  the  end  of  the  chord  nearest  S\  and 
X  and  Y  from  the  column  giving  the  other  end 
station. 

For  the  chords  from  S  to  the  other  stations  X9 
and  Y9  become  o,  and  (19)  reduces  to 

2\.                   A  f     x 

—  =  tang  A, (20) 

giving  the  angles  of  line  A. 
As  A  =  2N'2  in  minutes, 


-     A  t     \ 

=  —  in  degrees.      .     .     (21) 


N* 
Substituting  —  for  a  in  (9), 


D°n 


10  200 

and 


/ 
(22) 


D°n  =  2GW*  =  Z  of  table;  whence 
n  =      0     and  * 


THE    TRANSITION-CURVE    FIELD-BOOK.  II 

Substituting  for  n  its  value  Nc, 

.     Nc  —        0       and     c  —  ~J^~-    •     •     (23) 

As    the  Transition    Curve  o*f  N  stations  passes 
through  the  middle  of  the  adjusted  gap  q  at  station 

N  g 

-  its  ordinate  at  that  point  equals—;  or 

x  N  =  -     and     q  = 

.  2 


2C>N 

From  (23),  D°  —  -  — .     Therefore 


qX  D°  =  2XxX  —  =  —  *„,     .     (24) 

2  2 

making  c  =  100,  x  =  X ,  and  ^Z>°  =  0.4  NXN. 

2 

When  N '=  12,  ^Z>°  =  12.563  X  4.8  —  60.302. 

As  the  values  of  q  are  proportional  to  those  of  x, 
or  to  TV^3,  and  those  of  D°  to  TV^,  the  values  of 
q  X  -£*°  <irc  proportional  to  7V4.  Therefore 

i24  :  60.302  ::  N*  :  qD°, 
and 


qZ)Q  =  _  =  0.oo29o8i7V4.     .      (25) 

12 

Making  ^"=1,2,  3,  4,  etc.,  the  values  of  gD°  =  Q 
of  the  table  are  obtained;  whence 


12  THE    TRANSITION-CURVE    FIELD-BOOK. 

The  values  of  qD°  =  Q  at  the  even  stations 
tested  by  formula  (24)  result  in  Q9  =  0.0466, 
<24~  0.7446,  06=  3.770,  <28  =  ii. 914,  and  <210  =  29.o84, 
being  practically  the  values  given  in  the  table. 

x 
The  expression  q  =  -,  formula  (i),  in    which  x 

4 

represents  the  ordinate  to  the  cubic  parabola  at  Z, 
applies  approximately  to  the  Transition  Curve,  but 
not  nearly  enough  for  accurate  work  when  Z  is 
beyond  station  6.  The  actual  relation  between 
the  values  of  the  gap  q  and  the  end  ordinate  x  is 
given  in  line  F  of  the  table,  its  values  being  sim- 
ply those  of  q  in  decimal  parts  of  x  obtained  from 
the  division  of  q  by  x.  As  both  q  and  x  vary 
directly  in  proportion  to  the  chord  length,  the  ex- 
pression F  =  —  is  general,  and  in  all  cases 

q  =  Fx.  ......     (26) 

The  factor  F  is  used  in  determining  q  when  the 
offset/  is  fixed. 

EXPLANATION  OF  GENERAL  TABLE. 

All  of  the  quantities  in  each  column  of  the  table 
have  reference  to  the  station  of  that  column,  ex- 
cept that  the  quantity  Q  always  refers  to  a  point 
midway  between  the  station  of  its  column  and  £. 

Line  Ogives  the  station  numbers  of  the  Transi- 
tion Curve,  station  o  being  at  £. 

The  following  relations  exist  between  N  and 
other  quantities: 

Z  of  table; 


THE    TRANSITION-CURVE    FIELD-BOOK.  13 

N* 

—  =  a  in  degrees; 

2N*  =  A  in  minutes; 
O.2JV  =  D°  when  c  =  100; 
=  c  when  JD°  —  i. 


Line  X  gives  the  ordinates  from  the  tangent  SK 
to  the  stations  of  the  Transition  Curve  of  loo-foot 
chords. 

For  a  chord  c  of  any  other  length  the  station  or- 

dinate  x  =  X  X  -  . 
100 

The  values  of  X  in  the  table  are  the  sums  of  the 
products  of  the  consecutive  loo-foot  chords  mul- 
tiplied each  into  the  sine  of  its  inclination  to  the 
tangent  SK. 

X=  Ftang  J, 
x  =  Q.2NX     when 


Line  Y  gives  the  distances  on  the  tangent  SK 
from  S  to  the  station  ordinates  of  the  Transition 
Curve  of  loo-foot  chords. 

For  a  chord  c  of  any  other  length  the  ordinate 

distance  y  =  Y  X  --  . 
100 

The  values  of  Y  in  the  table  are  the  sums  of  the 
products  of  the  consecutive  loo-foot  chords,  multi- 
plied each  into  the  cosine  of  its  inclination  to  the 
tangent  SK. 

Y  —  X  cotang  A\ 

y  =  Q.2NY     when     D°  =  i. 


14  THE    TRANSITION    CURVE    FIELD-BOOK. 


Line  F  gives  the  values  of  q  at  -  in  decimal  parts 

of  x  at  n. 
Its  value  is 


x 

Line  Q  gives  at  each  station  N  the  gap  midway 
from  N  to  S,  between  the  tangent  SK  and  the 
parallel  tangent  of  a  i°  curve  connecting  with  the 
Transition  Curve  at  JV. 

For  any  other  degree  of  curve  the  gap  q  —  -^5. 
The  values  of  Q  in  the  table  are 
qD°  —  0.0029081^*; 

q  =  Q     when     D°  =  i. 

As  q  is  at  the  point  midway  between  Z  and  6*  it 
may  fall  at  any  whole  or  half  station  up  to  station 
6,  but  not  beyond. 

Line  Z  gives  the  lengths  of  Transition  Curves 
connecting  with  i°  curves  at  the  several  stations. 

2* 

For  any  other  degree  of  curve  the  length  n=-=-£. 

The  value  of  Z  in  the  table  is 


c  =  2ooa  = 
n  —  Z  when  D°  =  i. 

Line  C  gives  the  semi-chord  of  the  arc  of  a  i° 
circular  curve  of  which  a  subtends  half  that  arc,  — 
and  is  used  to  simplify  the  determination  of  the 
position  of  S,  the  initial  point  of  the  Transition 
Curve. 


THE    TRANSITION    CURVE    FIELD-BOOK.  15 

Its  value  is 

C=  573°  sin  »• 

For  a  circular  curve  of  other  than  i°  the  value 
of  C  is  to  be  divided  by  Z>°,  the  degree  of  curve,  in 
degrees  and  decimals. 

Line  a  gives  the  central  angles  turned  in  the 
Transition  Curve  between  tangent  SK  and  the 
sub-tangents  at  the  several  chord  stations. 

Its  value  in  degrees  and  decimals  is 

N*       cDGN        Z 
a  =  -  -  ; 

10  200  200 

versin  a  —  0.00002616^^. 

Line  A  gives  the  deflection  angles  between  tan- 
gent SK  and  the  chords  from  S  to  the  several 
chord  stations. 

Its  value  in  minutes  is 

A  =  2N*  =  --; 
10 

X 

=-. 

The  remaining-  lines  of  the  table  give  the  angles 
of  inclination  /of  the  chords  to  the  tangent  SK. 
each  line  giving  the  inclination  of  the  chords  from 
the  station  beginning  that  line  to  the  stations 
forward  of  it. 

X-  X' 
tang  /  =--  -ylT'y- 

Each  angle  in  line  A  and  those  following  it 
shows  the  inclination  of  a  chord  to  the  tangent  SK, 


l6  THE    TRANSITION-CURVE    FIELD-BOOK. 


O 


ooooooo       oo       oo 
»n    in  vO    O     r^   co     00     M     0»     en 


n  "^  "w  ^>  V  O  "<N  "N  O  "V  O  "<N  >» 
^00aSNO^-CNMOOO 
M000000000000 

O       cj       rtTtTflOOO       t^OO        OO       t-l 


in  "o    ^     oT    N     O  °°    ^  °^  ^  °^  ^  ^  ^°    ^  ^°    ^ 


M      M     <         r> 

S  JT  §  8' 


o 


oo     d     co    co    co    rt 


en 


O    co     O  Tf  co     -f-    -t  co    O    co 

^OjO-noooooo 


vQ 


O 


c? 


"    °M    °0    °0    °0 


^ 

Tt-    CO      O      CO 

1""!;^ 

o    o    o    o 


050000000^ 


THE    TRANSITION  -CURVE    FIELD-BOOK.  IJ 


EQUIVALENTS. 


Z      _  2oN*     _  2OOOf 

n  =  cJM  ==  —Q  =  -^-  ;     -^-; 


n       q  c  n 

cY       o.2NY      cX 

y  ==  -  -  —  — Frb—  = cotang 

100          D  100 


cX   •    0.2^^      rF  (9 


When   r  =  loo,  Z>°  =  0.2^,  «  =  loo^Y,  q  —  FX, 

=  X,y=Y. 

When   Z>°  =  i°,      ^==  20^,       n  =  Z,      $  =  Q, 
=  v.2NX'3  y  —  0.2NY. 


20  THE    TRANSITION-CURVE    FIELD-BOOK. 

In  this  case  TV=  5,  and 

*  =  S  =  2°-s  =  2°  30'. 

Observe  that,  in  using  the  table,  if  TV  and  c  are 
given,  the  other  elements,  x,  y,  ny  D°,  and  ^,  are 
readily  obtained.  Thus,  take  from  column  TV  the 
quantities  X,  Y,  Q,  and  Z.  Then  from  equivalents, 

x  =  Xx  —  ,y  =  YX— ,£#=*, -=D°,-Q=4. 

ioo  ioo  n  D 

The  angles  a  and  A  are  also  taken  from  column  TV. 

The  elements  include  D°  because  it  is  the  rate 
of  curvature  of  the  Transition  Curve  at  the  junc- 
tion Z,  as  well  as  the  degree  of  the  circular  curve. 

If  in  addition  to  TV  some  other  element  than  c — 
as  x,  y,  /i,  D  ',  or  q — is  given,  from  equivalents, 

ioo  ioo         n  Z       _  2&Nq 

-x  X    =  y~Y   :=  W=I?W=    ~Q~"' 

and  with  N  and  c  known,  the  other  elements  are 
obtained  as  above. 

If  a  value  of  the  central  angle  a  in  one  of  the 
columns  of  the  table  is  selected  and  any  other 
element  is  known,  the  column  containing  the 
selected  value  of  a  fixes  TV,  and  the  remaining  ele- 
ments are  determined  as  in  the  last  case. 

Methods  of  determining  the  elements  with  other 
data  than  those  above  mentioned  are  shown  in  the 
Problems. 

DEFINITIONS  AND  NOTES  REFERRING  TO  PROBLEMS 

FOLLOWING. 

»£  =  Initial  station  of  Transition  Curve. 
.    L  =  Point  of  junction  of  Transition  and  Circular 
Curves. 


THE    TRANSITION-CURVE    FIELD-BOOK.  21 

/—  Angle  of  intersection  of  tangents  to  be  con- 
nected by  Transition  and  Circular  Curves. 
75*  =  Distance  on  tangent  from  S  to  intersection 

point. 
P  —  Point   on  IS  opposite  the  PC  (or  P.T.)  of 

original  Circular  Curve. 
SP  =  Distance  on  AS*  from  S  to  P. 
SK  —  Distance  on  tangent  from  S  to  point  K,  op- 
posite L. 

p  =  Unadjusted  offset  between  given  tangent 
and  parallel  tangent  of  circular  curve  to 
be  connected  with. 

q  =  Adjusted  gap  between  tangent  SK  and  par- 
allel tangent  of  connecting  circular  curve. 
6  =  Angle  turned  in  arc    from  P.C.C.  to  P  when 

circular  curve  is  compounded. 
The  radius   corresponding   to  D°  is  denoted  by 
R;  to  Z>'°  by  ./?';  to  7>"°  by  .tf". 

When  n,  q,  or  other  quantity  is  assumed  approx- 
imately, this  is  expressed  by  na,  qa,  etc. 

When  such  expressions  as  xL9  XN,  K10,  etc.,  are 
used  L,  N,  I0,  etc.,  indicate  stations  to  which  the 
larger  letters  refer. 

The  central  angle  a  cannot  exceed  \I,  and  un- 
less 7  is  very  small  it  is  better  that  it  should  not 
much  exceed  -J-7. 

When  the  degree  of  the  connecting  curve  is  less 
than  5°,  by  limiting  the  value  of  q  to  about  2  feet 
per  degree  of  curve  the  length  of  the  Transition 
Curve  may  be  limited  to  about  500  feet.  In  any 

,  60.302 
case  q  must  not  exceed  — =-  . 

All  lengths  are  expressed  in  feet  and  decimals 
of  a  foot. 


20  THE    TRANSITION-CURVE    FIELD-BOOK. 

In  this  case  IV  =  $,  and 

a  =  ^  =  2°.5  =  2°  30'. 

10 

Observe  that,  in  using  the  table,  if  N  and  c  are 
given,  the  other  elements,  x,  y,  n,  Z>°,  and  q,  are 
readily  obtained.  Thus,  take  from  column  N  the 
quantities  X,  Y,  Q,  and  Z.  Then  from  equivalents, 


The  angles  a  and  A  are  also  taken  from  column  N. 

The  elements  include  D°  because  it  is  the  rate 
of  curvature  of  the  Transition  Curve  at  the  junc- 
tion Z,  as  well  as  the  degree  of  the  circular  curve. 

If  in  addition  to  N  some  other  element  than  c  — 
as  xy  y,  //,  jO\  or  q  —  is  given,  from  equivalents, 

100  100  _     n  Z       _  2oJVq 

=  *—  =  y~y  :=  jv  =  WN^  ~Q~' 

and  with  A7  and  c  known,  the  other  elements  are 
obtained  as  above. 

If  a  value  of  the  central  angle  a  in  one  of  the 
columns  of  the  table  is  selected  and  any  other 
element  is  known,  the  column  containing  the 
selected  value  of  a  fixes  N,  and  the  remaining  ele- 
ments are  determined  as  in  the  last  case. 

Methods  of  determining  the  elements  with  other 
data  than  those  above  mentioned  are  shown  in  the 
Problems. 

DEFINITIONS  AND  NOTES  REFERRING  TO  PROBLEMS 

FOLLOWING. 

S  =  Initial  station  of  Transition  Curve. 
.    L  =  Point  of  junction  of  Transition  and  Circular 
Curves. 


THE    TRANSITION-CURVE    FIELD-BOOK.  21 

I '  —  Angle  of  intersection  of  tangents  to  be  con- 
nected by  Transition  and  Circular  Curves. 
fS=  Distance  on  tangent  from  S  to  intersection 

point. 
P  —  Point   on  IS  opposite  the  PC  (or  P.T.)  of 

original  Circular  Curve. 
SP  -  Distance  on  /£  from  S  to  P. 
S K  =  Distance  on  tangent  from  S  to  point  K,  op- 
posite L. 

p  =  Unadjusted  offset  between  given  tangent 
and  parallel  tangent  of  circular  curve  to 
be  connected  with. 

q  =  Adjusted  gap  between  tangent  SK  and  par- 
allel tangent  of  connecting  circular  curve. 
0  =  Angle  turned  in  arc    from  P.C.C.  to  P  when 

circular  curve  is  compounded. 
The  radius   corresponding   to  U°  is  denoted  by 
J?;  to  Z>'°  by  R'\  to  Z>"°  by  R" . 

When  /z,  q,  or  other  quantity  is  assumed  approx- 
imately, this  is  expressed  by  na,  qa,  etc. 

When  such  expressions  as  xLy  XN,  K10,  etc.,  are 
used  £>AT>  I0,  etc.,  indicate  stations  to  which  the 
larger  letters  refer. 

The  central  angle  a  cannot  exceed  |/,  and  un- 
less /is  very  small  it  is  better  that  it  should  not 
much  exceed  ^/. 

When  the  degree  of  the  connecting  curve  is  less 
than  5°,  by  limiting  the  value  of  q  to  about  2  feet 
per  degree  of  curve  the  length  of  the  Transition 
Curve  may  be  limited  to  about  500  feet.  In  any 

,  60.302 
case  q  must  not  exceed  — =- . 

All  lengths  are  expressed  in  feet  and  decimals 
of  a  foot. 


22  THE    TRANSITION-CURVE    FIELD-BOOK. 

PROBLEM  I. 

Transition  Curve  to  be  laid  off  by  deflection  angles. 
Required  field-work  and  record  of  field-notes. 

Given  initial  point  S,  chord  length  c,  and  num- 
ber of  chords  N. 

EXAMPLE,  c  =  25,  N  —  10. 

CASE  i.  Curve  to  be  run  from  S  to  L. 

Set  the  instrument  over  S,  and  if  all  the  stations 
can  be  seen  from  that  point,  take  from  line  A  of 
table  the  deflection  angles  2',  8',  18',  32',  50',  i°  12'^ 
i°  38',  2°  08',  2°  42',  and  3°  20',  and  with  vernier  at 
o  on  tangent  SK  deflect  these  angles  successively 
for  stations  i  up  to  10;  irsing  the  given  chord 
length,  25  feet,  between  eacli  two  consecutive  sta- 
tions. 

Move  to  station  10,  and  with  vernier  at  3°  20' 
sight  on  S  and  turn  to  10°,  line  a,  giving  the  com- 
mon tangent  to  Transition  and  Circular  Curve, 
whence  the  latter  is  run  as  usual. 

In  this  case 


0  _   Z        2000 

X/   v,   -  


25  x  10 

The  field-notes  areas  follows,  reading  from  the 
bottom    upwards.     S  is  supposed   to   fall  at  R.R. 
station  210  -(-  24.     The  distance  column  shows  the 
total    distances    from    S    to    the    several    stations' 
measured  by  consecutive  chords. 

L  212  +  74  (10)     250     3°  20'     io°.oo 

(9)   225   2%7' 
(8)   200   2°  08' 

(7)  175  i°38' 
(6)  150  i°  12' 


THE    TRANSITION-CURVE    FIELD-BOOK. 


(5) 

125 

So' 

(4) 

IOO 

32' 

(3) 

75 

18' 

(') 

5° 

8' 

(0 

25 

2' 

(o) 

0 

o' 

3°.  20 

•5*210  + 

With  instrument  at  (10)  and  vernier  at  3°  20' 
sighted  on  S,  by  turning  successively  to  3°  42', 
4°  08',  4°  38',  etc.,  (column  10  of  table,)  the  points 
set  at  i,  2,  3,  4,  etc.,  will  be  tested. 

CASE  2.  Instrument  required  to  be  set  at  inter- 
mediate stations  between  S  and  Z,  as  (3)  and  (7). 

Set  over  S  and  deflect  from  SK  the  angles  2',  8', 
and  18',  line  A,  for  stations  (i),  (2),  and  (3). 

Move  to  (3),  and  with  vernier  at  18'  sight  on  S, 
and  turn  to  1°  14',  i°  38',  2°  06',  and  2°  38'  (line  (3), 
table)  for  stations  (4),  (5)^  (6),  and  (7). 

Move  to  (7),  and  with  vernier  at  2°  38'  sight  on 
(3),  and  turn  to  5°  38',  6°  26',  and  7°  18'  (line  (7), 
table)  for  stations  (8),  (9),  and  (10). 

Move  to  (10),  and  with  vernier  at  7°  18'  sight  on 
(7),  and  turn  to  10°,  line  /r,  giving  common  tangent 
at  (10)  or  L  as  before. 

The  field-notes  are: 

L  212  -f-  74  (io)O    250     7°  18"      io°.oo 


S  210  -j-  24 


(9)  225 

(8)     200 

(7)0  175 
(6)   150 
(5)   125 
(4)   loo 
(3)0  75 
(2)    50 

(0   25 

(o)O   o 

6°  36' 

5°  38' 
2°  38' 

7°  18' 

2°  06' 

i°38' 
i°i4' 
18' 

2°  38' 

8' 

2' 
o 

18' 

24  THE    TRANSITION-CURVE    FIELD-I3OOK. 

The  vernier  readings  repeated  as  backsights  are 
underscored. 

CASE  3.  Curve  run  backward  from  Z  to  S.  Same 
data  as  before,  except  that  Z  is  given. 

Set  over  Z  =  (10),  and  turn  into  tangent  to  the 
8°  curve. 

With  vernier  at  10°  and  instrument  on  tangent 
at  Z,  turn  to  9°  02',  8°  08',  and  7°  iS'  (column  (10), 
table),  for  stations  (9),  (8),  and  (7). 

Move  to  (7),  and  with  vernier  at  7°  18'  sight  on 
(ro)  or  Z,  and  turn  to  4°  14',  3°  38',  3°  06',  and 
2°  38'  (column  (7),  table)  for  stations  (6),  (5),  (4), 
and  (3). 

Move  to  (3),  and  with  vernier  at  2°  38'  sight  on 
(7),  and  turn  to  38',  26',  and  18'  (column  (3),  table) 
for  stations  (2),  (i),  and  S. 

Move  to  S,  and  with  vernier  at  18'  sight  on  (3), 
and  turn  to  o,  giving  tangent  SK. 

The  field-notes  are: 

S  212  +  74     (o)O   250          ij^    o°    o' 


(0 

225 

26' 

w 

200 

38' 

(3)0 

J75 

2°  38' 

1  8' 

(4) 

150 

3°  06' 

(5) 

I25 

3°  38' 

(6) 

IOO 

4°  14' 

(7)0 

75 

7°  18' 

2°  38' 

(8) 

5° 

8°  08' 

(9) 

25 

9°  02' 

24  (io)Q 

o 

o    ^ 

10  oo 

7°  18' 

Z  210 

Whether  the  Transition  Curve  is  run  from  S  to  Z 
or  from  Z  to  £,  the  notes  given  are  to  be  read  from 
the  bottom  upward. 

Observe,  generally,    that    when    the    Transition 


THE    TRANSITION-CURVE    FIELD-BOOK.  2$ 

Curve  is  to  be  run  from  ^S*  to  Z,  the  deflection  an- 
gles are  taken  from  the  lines  of  the  instrument 
stations  in  the  table;  and  when  from  L  to  £  they 
are  taken  from  the  columns  of  the  instrument  sta- 
tions. Or,  in  other  words,  when  the  station  to  be 
sighted  on  is  between  the  instrument  station  and 
L  or  at  Z,  the  corresponding  vernier  angle  is  in  the 
line  of  the  instrument  station;  and  when  the  sta- 
tion to  be  sighted  on  is  between  the  instrument 
station  and  S  or  at  S,  the  corresponding  vernier 
angle  is  in  the  column  of  the  instrument  station. 

In  all  cases  the  vernier  angle  for  turning  into 
tangent  is  in  line  a  and  in  column  of  instrument 
station. 

If  the  chord  length  is  given  and  any  two  station 
points  of  the  Transition  Curve  are  in  position  on 
the  ground,  the  other  stations  can  be  readily  re- 
set. For  instance,  in  a  i2-chord  Transition  Curve 
with  chord  length  known,  and  stations  (4)  and  (9) 
in  place,  when  the  instrument  is  set  over  (4),  with 
vernier  at  4°  26'  (line  (4)  column  (9)  ),  sight  on  (9), 
and  for  the  stations  from  (5)  to  (12)  turn  to  the 
corresponding  angles  from  2°  02'  to  6°  56.'  of  line 
(4),  and  for  stations  (3)  to  (S)  turn  to  the  corre- 
sponding angles  from  i°  14'  to  o°  32'  of  column  (4). 
If  the  instrument  is  set  at  (9),  with  vernier  at  4°  26' 
as  before,  sight  on  (4),  and  for  the  stations  from  (9) 
to  (12)  turn  to  the  corresponding  angles  from 
9°  02'  to  11°  06'  of  line  (9),  and  for  stations  (8)  to 
£  tlirn  to  the  corresponding  angles  from  7°  14'  to 
2°  42'  of  column  (9). 

To  turn  into  tangent  at  any  point,  with  vernier 
reading  the  angle  of  inclination  of  any  chord  end- 
ing at  the  instrument  station,  sight  on  the  station 
point  at  the  other  end  of  that  chord,  and  turn  the 


26  THE    TRANSITION-CURVE    FIELD-BOOK. 

vernier  to  the  angle  of  line  a  in  the  column  of  the 
instrument  station.  Thus,  with  instrument  at  (9), 
sight  on  any  station,  as  (4),  with  vernier  at  4°  26', 
(or  on  (2)  with  vernier  3°  26',  or  on  (12)  with  ver- 
neir  11°  06'),  and  turn  vernier  to  8°  06'  for  tangent. 

When  the  instrument  is  at  Sy  sight  on  any  station 
with  vernier  at  the  corresponding  angle  in  line  //, 
and  turn  to  o  for  tangent;  when  the  instrument  is 
at  Z,  sight  on  any  station  with  vernier  at  the  cor- 
responding angle  in  column  Ly  and  turn  to  angle  a 
in  column  L  for  tangent. 

Thus  with  instrument  at  S,  sight  on  (10)  with 
vernier  3°  20'  (or  (i  i)  with  vernier  4°  02'),  and  turn 
to  o  for  tangent.  With  instrument  at  L  =  (10), 
sight  on  Jy  with  vernier  3°  20'  (or  on  (3)  with  ver- 
nier 4°  38'),  and  turn  to  10°  for  tangent. 

In  running  the  Transition  Curve,  the  vernier 
reading  used  in  setting  the  station  moved  to  is 
always  the  vernier  reading  to  be  used  for  back- 
sight on  the  station  moved  from. 

As  all  the  angles  in  the  table  represent  the  in- 
clination of  chords  (or  sub-tangents  in  line  a)  to 
the  tangent  SK,  the  field  test  of  the  correctness  of 
any  angle  turned  in  accordance  with  the  foregoing 
instructions  is  as  follows: 

When  the  instrument  with  its  proper  vernier 
reading  is  sighted  on  a  station  and  the  vernier  is 
then  turned  to  o,  the  telescope  will  be  parallel  to 
tangent  Sl£,  and  the  needle  will  give  its  bearing. 

This  will  in  all  cases  give  a  practical  answer  to 
the  question  whether  to  set  the  vernier  to  the  right 
or  to  the  left  of  o,  because  when  the  instrument  is 
sighted  on  a  station  with  the  vernier  properly  set, 
and  the  vernier  then  turned  to  o,  the  needle  will 
give  the  bearing  of  tangent  SK,  whilst  if  set  on 


THE    TRANSITION-CURVE    FIELD-BOOK.  27 

the  wrong  side  of  o,  the  needle  will  show  the  error 
doubled. 

This  test  is  of  course  applicable  at  any  station 
whilst  running  in  the  Transition  Curve,  and  by  its 
use  angular  errors,  if  there  are  any,  are  readily 
found  and  corrected. 

PROBLEM  IT. 

To  lay  off  the  Transition  Curve  by  ordinates. 
Given    initial    point    S  on    tangent    SK,    chord 
length  c,  and  number  of  stations  N. 

EXAMPLE,  c  =  25,7V^=:io.     The  ordinate  lengths, 

JTX  — i  for  stations  (i),  (2),   (3),  etc.,  are  ^-^  , 
100  4 

.4564    1.571 

— ,  —  -  ,  etc.,  —  .015,  .12,  .39,  .93,  1.82,  3.14,4.99, 
4  4 

7.44,  10.59,  and  14.51.     Similarly  the  ordinate  dis- 
tances frofn  *S  measured  on   tangent  SK  are  Y  X 

~  -  25>  5°>  75>  99-99.  124.98,  149.94,  174.87,  i99-75> 

224.55,  and  249.24. 

Measure  the  ordinate  distances  from  S,  and  set 
off  the  corresponding  ordinate  lengths  at  right  an- 
gles to  SK.  The  station  points  should  be  tested 
by  consecutive  chord  measurements. 

The  rule  for  laying  off  the  Transition  Curve  by 
ordinates  applies  to  all  cases  when  S,  c,  and  N 
are  determined.  The  ordinate  lengths  and  dis- 
tances vary  with  every  change  in  the  chord  length, 
but  are  a  Iways  obtained  as  in  the  example  just 
given.* 

*  In  the  problems  following,  the  solutions  are  given  for 
locating  by  deflection  angles,  but  after  determining  c,  S,  and  N, 
the  Transition  Curves  in  all  cases  may  be  laid  off  by  ordinates 
as  in  Problem  II, 


28  THE    TRANSITION-CURVE    FIELD-BOOK. 

PROBLEM  III. 

The  angle  of  intersection  I  of  two  tangents  being  given, 
to  connect  them  with  a  circular  curve  of  any  given 
degree  of  curvature  D°  by  Transition  Curves  having 
a  common  tangent  and  the  same  rate  of  curvature 
D°  at  their  points  of  junction  L  and  L'  with  the  cir- 
cular curve. 

The  example  in  each  of  the  five  cases  following 
gives  the  elements  of  only  one  of  the  two  Transi- 
tion Curves,  but  applies  to  both. 

In  figure  2, 

IS-IP-PK+KS. 
But 

IP  =  OP  tang  i/  =  (OB  +  BP)  tang  £7 
tang  |7; 


KS  =  y  -  Yx  —  \ 
100 


and  by  substitution  in  the  first  equation, 

tang  i/+         --5.       .     (27) 


In  formula  (27),  which  applies  to  all  the  cases  of 
this  problem    (but  not  in  the    formulas    of    other 

problems   following),  --    may    be    substituted    for 


THE    TRANSITION-CURVE    FIELD-BOOK.  29 


30  THE    TRANSITION-CURVE    FIELD-BOOK. 

CASE  i.  Given  the  degree  of  circular  curve  Z>°, 
the  number  of  stations  N,  and  the  intersection  an- 
gle/. 

Rule.  Take  from  column  N  of  table  the  quanti- 
ties Y,  <2,  Z,  and  C. 

Z  Q 

Then  from  equivalents  j^-^=.—cy  Nc  =  n,  -~0=^. 

Find  fS  by  (27),  and  lay  off  the  Transition  Curve 
as  in  Problem  I  or  II. 

EXAMPLE.  Z>°  =  8°,  N  =  12,  /—  50°. 

From  column  .A7^  of  table,  F12  =  1192.47,  Q^  ~ 
60.302,  Z12  =  2880,  C12  =  1424.9- 

Z  2880 

-  =  c  —  30;     30  X  12  =  n  —360; 


8  X  12 


IS  =  (X  +  f)  tang  K+         - 

=  (7I6.8  +  7.5)0.4663! 


=  337-75  +  357-75  -  J78.i  =  51  7-4- 

Measure  517.4  from  /  to  S  and  Sf  on  the  two 
tangents.  From  6*  lay  off  the  Transition  Curve 
by  angles  (Problem  I)  or  by  ordinates  (Problem  II) 
with  chord  length  30.  From  L  (sta.  N^  the  8° 
curve  is  to  be  run  in  for  the  angle  50°  —  2a  —  21° 
14'  to  Lf  of  the  second  Transition  Curve,  which 
is  to  be  run  backward  to  S'  on  the  second  tangent  : 

Or  both  Transition  Curves  may  be  run  forward 
from  S  and  5',  and  the  common  tangents  at  L  and 


THE    TRANSITION-CURVE    FIELD-BOOK.  31 

Z'  run  to  intersection  before  putting  in  the  21°  14' 
of  8°  curve. 

If  the  Transition  Curves  are  located  by  ordi- 
nates,  in  order  to  get  the  common  tangent  at  Z 
set  the  instrument  over  Z,  and  with  the  vernier  at 
Z/12  sight  on  S  and  turn  to  #12  for  the  tangent  at 
Z:  or  if  S  cannot  be  seen  from  Z,  set  the  vernier  at 
/  for  any  long  chord  ending  at  Z,  sight  on  the  sta- 
tion at  the  other  end  of  the  long  chord  and  turn 
to  a12.  In  the  present  case,  for  example,  if  S  can 
be  seen  from  Z,  set  the  vernier  at  4°  48',  sight  on  S 
and  turn  to  14°  24'  for  the  common  tangent:  or  if 
(4)  is  the  nearest  station  to  S  that  can  be  seen,  set 
the  vernier  at  /  for  chord  (4)  (12),  or  at  6°  56', 
sight  on  (4),  and  turn  to  14°  24'  for  tangent.  To 
get  common  tangent  at  L' ,  substitute  Z'  and  S' 
for  Z  and  S  in  above  description. 

Similar  field-work  is  required  in  all  of  the  cases 
of  Problem  III. 

A  very  convenient  test  of  the  arithmetical  values 
of  c  and  q  as  calculated,  in  all  of  the  problems,  is 

cNQ 

— ~  —  A  in  minutes.     In  the  present  case 

30  X  12  X  60.302  _  _ 

~Jo<TS8~~        12 

CASE  2.  Given  Z>°,  7,  and  approximate  length  of 
Transition  Curve  na. 

Rule.  Take  from  the  table  the  value  of  Z  nearest 
to  Z>°  X  na,  and  from  the  same  column  JV,  Q,  C, 
and  Y. 

Then  from  equivalents  -^  =  «,—-  =  r,  -^  =  q. 
For  the  rest  proceed  as  in  Case  i. 


32  THE    TRANSITION-CURVE    FIELD-BOOK. 

EXAMPLE.  £>°  =  8,  na  —  300,  7=  42°,  D°  X  na  =  8 
X  300  =  2400,  and  nearest  Z=  2420  from  column 
7Vn.  Takeout  (?11=42-597>  Kn  —  1095.13,  C=i2oi.o. 


2420  302.5 

--  =  «  =302.5;  -=f  = 


(9        42. 

—    =      — 
7~)° 
JLS 


is  =  (1146.3  +  5-3)  -38386  +  ?H^~~^: 
=  442.05  +  301.16  —  240.2  =  503.0. 

CASE  3.  Given  Z>°,  /,  and  a  assumed  as  equal  to 
one  of  its  values  in  the  table.  If  /  is  less  than 
about  40°,  assume  a  between  ^/and  |7. 

Rule.  Find  the  desired  value  of  a  in  the  table, 
and  from  the  same  column  take  out  the  quantities 
Z,  Q,  C,  and  F,  and  proceed  as  in  Case  2. 

EXAMPLE.  D°  —  5,  /—  20°  20',  and  a  from  col- 
umn JV6  —  6°  24'. 

Zs  =  1280,  QB  =  11.912,  C6  =  638.7,  F8  =  799.01; 
1280  256  11.912 

-_  =  «  =  256;   -|-  =  c  =  32;   -y-  ==,?  =  2.382, 


IS  =  (1146.3  +  2.4)0.17933  +  799  X  0.32  ~  — 
=  206.0  +  255.7  -  127.7  =  334.0. 


CASE  4.  Given  Z>°,  7,  and    approximate  gap   < 
Note  that  qa  x  Z^°  must  not  exceed  about  60.302. 


THE    TRANSITION-CURVE    FIELD-BOOK.  33 

Rule.  Take  from  table  the  value  of  Q  nearest  to 
qa  X  -Z?0,  and  from  same  column  Z,  C,  and  Y.  Pro- 
ceed as  before. 

EXAMPLE.  D°~ 20°,  7  =50°,  qa  =  1.5,  1.5X20=30. 
Nearest  Qirom  column  ^^  =  29.081;  and  Z10  = 
C*,o  =  994-9>  F!o  =  996-97- 


2Q.o8l  2000  100 

—  q  =  1.454; =  «  =  100;  - —  =  <;  =  10; 

20  y  °^       20  10 


IS=  (287.9  +  i-S)  i-WIS  +  997  X  o.i  - 
=  344.8  +  99.7  —  49.7  =  394.8. 

CASE  5.  Given  /,  N,  and  chord  length  c.  In  this 
case  D°  cannot  be  assumed. 

Rule.  From  column. TV"  take  Q,  Z,  C,  and  F,  and 
from  equivalents, 


Proceed  as  before. 

EXAMPLE.  Given  7=40,  C—  30,  N —  9.  From 
column  9  of  table  (^=.19.080,  Za  =  1620,  C9  =  807.3, 
Fa  =  808.22. 


30  X  9  ^  *  =  no;  =  ^  =  6*;  =fe=i«8; 

7^  =  (955.4  +  3.2)  0.36397  +  898.2  X  0.3  -  5^ 
—  348.9  +  269.5  — 


34 


THE    TRANSITION-CURVE    FIELD-BOOK. 


PROBLEM  IV. 

To  locate  the  Transition  Curve  when  both  circular  curve 
and  tangent  are  fixed  in  position,  and  the   offset  p 

60 
is   less   than  about  — -^. 

Given  /,  £)°,  and/. 

In  the  figure,  R  —  the  radius  of  the  given  curve, 
and  BP  =  the  given  offset/.  Also  LK  =  the  end 
ordinate  x,  and  JB'P'  =  gap  q. 


eo 


THE    TRANSITION-CURVE    FIELU-BOOK.  35 

LK  -  x  —  MB  +  BP  —  R  versin  a 


and 

q  —  Fx  —  F(R  versin  «+/).     .     .     .     (28) 

SP  =  SK-KP; 

SK=—\ 

100 

Therefore 


Rule.  Take  from  table  the  value  of  Q  next  lower 
than  p  X  Z>°,  and  from  same  column  Z,  jF,  F,  C, 
and  a:. 

Find  ^  by  (28). 


Find  5^  by  (29). 

EXAMPLE.  /  =  36,  D°  =  6,  /  =  9.  Then/Z>°  = 
54,  and  next  lower  value  of  Q  (from  table)  is  (?n  = 
42.577;  and  Zn  —  2420,  F^  =  .25072,  F^  1095.13, 

C  =    I2OI.O,   ^  =    12°   06'. 

?  =  •15°72(955-4  X  .022217  +  9)  =  7.578; 


2420  430.72 

=  «  =  430.72;  --  =  ,=  39.16; 


sp  _  39-16  X  1095.  13  _  I20I.O 
100  6 

—  428.85  —  200.17  =  228.68, 


36  THE    TRANSITION-CURVE    FIELD-BOOK. 

At  L  the  Transition  Curve  rate  is  5°  37',  and  the 
circular  curve  rate  6°. 

NOTE.  —  In  cases  where/  X  D°  is  nearly  equal  to  the  nearest 
value  of  Q,  that  and  the  corresponding  table  quantities  should 
be  taken  even  when  Q  is  greater  than  /  X  £>°.  The  resulting 
rate  of  curvature  of  the  Transition  Curve  at  L  will  be  some- 
what greater  than  that  of  the  connecting  circular  curve,  and 
will  call  for  a  greater  elevation  of  the  outer  rail;  but  this  differ- 
ence is  too  slight  to  be  of  any  practical  importance.  On  the 
other  hand,  the  actual  difference  between  the  rates  of  curvature 
of  the  two  curves  at  L  will  be  less,  in  such  cases,  than  when 
the  next  lower  value  of  Q  and  the  corresponding  table  quanti- 
ties are  taken. 

SECOND  EXAMPLE  under  Problem  IV  (see  pre- 
ceding note). 

7=  36°,  D°  =  6,  /  =  9,  fJD°  =  54  as  before. 
The  nearest  value  of  Q  is  Q12  =  60.302;  and  F^  = 
.25105,  Z12  =  2880,  <7ia  =  1424-9)  YH  =  ii92-47, 
al2  =  14°  24';  and  from  (28), 

q  —  F(R  versin  a  +/) 

=-25I05(955-4  X  .03142  +  9)  =  9-79S6; 
=  ZX°  =  6°  .156-6°  en*'; 


- 

9.7956 

2880  476.83 

6^6-==*  =  467-83;^  =;  =  38.99; 

on  _  38.99  X  1192.47    I424-9 
o./  —  —    — 

ioo          6 

=  464-94  —  237.48  —  227.46. 

At  Z  the  Transition  Curve  rate  is  6°  09^'  and  the 
circular  curve  rate  6°. 


THE    TRANSITION-CURVE    FIELD-BOOK.  37 

PROBLEM  V. 

To  locate  the  Transition  Curve  when  both  circular  curve 
and  tangent  are  fixed  in  position  and  the  offset  p  is 

60 
greater  than  about  —  -$. 

Given  /,  Z>°  and/. 

In  Fig.  4,  B'P9  =/;  BP  =  /;  7?  =  radius  of 
given  curve  Z>°;  and  0=  central  angle  from  P.C.C. 
to  point  of  tangent  parallel  to  SK. 

M'B'  -  MB  =  BP  -  B'P'  =  p  -  /. 

But  M'B'  =  R'  versin  8  and  MB  —  R  versin  ft 
Therefore  (^'  —  7?)  versin  0  =/  —/  and 

.  _  ,f 
versin  0  =  J>T^T^>  .....     (3°) 

A:  =  LK  =  R  versin  a 


and 

^  —  Fx  =  (J?  versin  «  +p')F.  .     •     .     (31) 


SP=  SX-  KP'  +  P'K'  -  K'P 

e'  •  •  (32) 


.   Assume  Z>'°  about  o.gZ>°  and/'  not  greater 

60 
than  about  -jyt>- 

Find  #  from  (30). 

Take  from    column  giving  Q   next  lower  than 
p'  X  D'Q  the  quantities  <?,  Z,  C,  F,  ^,  and  a. 
Find  ^  from  (31). 

Q  _    n,,o       _Z_  n  _ 

q   ~  '      i>'/5~      '      ~N~ 


THE    TRANSITION-CURVE    FIELD-BOOK. 


Find  SP  from  (32)0 


THE    TRANSITION-CURVE    FIELD-BOOK.  39 

EXAMPLE.  /=  60°,  Z>°  =  8°,/  =  16. 
Assume  Z>'°  =  7°;  /'  =  9. 
From  (30) 

-— — &- —  =r =  0.06846  =  versin  0 

819  02  —  716.24       102.24 

—  versin  21°  19'. 


7  X  9  =  63-     From  TV^  ,  £19  =  60.302,  Zia  P?  2880, 
<7I9  =  1424-9.   ^,a 


14°  24'. 


(819.02  X  0.03142  -4-  9)  0.25105  =  q  =•  8.72. 


8.72 


2*880  416.5 

_  =  «  =  4i6.S;—  =  *  =  34-71. 


SP  =  *    ^^2  ^  +(8i9.o2-7i6.24).3635 

=  413-92  —  203.56  +  37.16  =  247.5. 

Measure  247.5  ^eet  from  P  and  put  in  S,  Run  in 
Transition  Curve  from  6"  with  12  chords  each  34.71 
long.  From  L  (station  12)  run  7°  curve  for  21°  19' 
—  14°  24'  =  6°  55'  to  P.C.C.y  there  connecting  with 
the  8°  curve. 

The  point  S  may  be  determined  on  the  ground 
without  calculating  SP,  by  running  in  the  6°  55' 
of  7°  curve  from  P.  C.  C.  to  Z;  turning  off  90°  —  a 
outwardly  from  the  tangent  at  Z  and  intersecting 


40  THE    TRANSITION-CURVE    FIELD-BOOK. 

cY 
SK  at  K\  and  thence  measuring  -  -  =  413.92  to  S: 

TOO 

or  the  Transition  Curve  may  be  run  in  backward 
from  L  without  first  determining  the  position  of  S. 
When  convenient,  however,  it  is  better  to  fix  both 
vS*and  L  before  running  in  SL. 

In  this  example  the  rate  of  curvature  of  the 
Transition  Curve  is  6°  55'  at  Z,  its  junction  with 
the  7°  curve. 

The  difference  in  the  rates  of  the  Transition  and 
Circular  Curves  at  L  need  not  absolutely  be  less 
than  at  other  P  C.C.'s  of  the  line;  but  in  order 
properly  to  adjust  the  superelevation  of  the  outer 
rail,  the  differences  in  the  rates  of  curvature  at  all 
points  of  compound  curve  should  be  as  small  as 
circumstances  will  permit. 

Problem  V  is  limited  by  the  conditions  that  0 
must  be  less  than  |7  and  greater  than  a.  If  6  as 
calculated  is  greater  than  -J-/,  increase  either  the 
assumed  value  of/'  or  of  the  radius  of  D'°\  and  if 
the  calculated  6  is  Jess  than  a,  diminish  one  or  the 
other  of  these  assumed  quantities. 

It  is  always  practicable,  and  in  some  cases  pref- 
erable, to  find  the  external  distance  HI  from  the 
middle  of  the  curve  D°  to  the  intersection  of  tan- 
gents (HI  —  (R  -{-p)  exsec  £/  +  /),  and  substitute 
Problem  VII  for  Problem  V. 

PROBLEM  VI. 

Line  to  be  held  over  the  middle  portion  of  a  circular 
curve  of  large  central  angle  connecting  two  tangents, 
to  shift  curve  inwards  at  the  ends  and  put  in  Transi- 
tion Curves. 

Given  JD°  and  point  of  curve  (or  tangent)  P. 
In  the  figure,  BP'  =  q. 


THE    TRANSITION-CURVE    FIELD-BOOK. 


01 


q  =  MP  -  M'B  =  R  versin  0  -  R'  versin  0, 


and 


versin  (9  =  ^4»  ....     (33) 


SP  =  SK  -  KP'  +  P'K'  -  K'P, 
SK  =  — ,    ^^'  F=  7^6,    /"A7  =  ^'    sin    6>,    and 

I  CO  JJ 

K'P  =  R  sin  ft 


42  THE    TRANSITION-CURVE    FIELD-BOOK. 

Therefore 

SP  =  C-^Q-~-(R-R')sm6.    .     (34) 

Rule.  Assume  the  curve  D'°  at  the  ends  about 
i.i  Z>°,  and  qa  =  4  feet  or  less.  Take  from  table  Q 
nearest  to  qa  X  Z)'°,  and  from  same  column  Z,  C, 
and  Y. 


Find  ff  by  (33),  and  SP  by  (34). 

EXAMPLE.  Z>°  =  8,  ^  —  3. 

Assume  D'°  —  8°.8;  3  X  8.8  —  26.4. 

From  column  1V10,  Q  =  29,081;  and  Z10  =  2000, 

,o  =  994-9.     ^,o  =  996.97. 


20.081 

-- 


From  (33) 


^      ^  :  - •=  versin  6  =  0.05077  =  versin  18°  20'; 

/       *  D    *  / 

2000  227.3 

—  =«  =  227.3;  -—  =  <  =  22.73. 
From  (34) 

a>  =  =Ji^H-^-«>.X  0.3-454 

=       226.6       —       H3.I        —       20.5       =       93-0. 

From  StoP.C.C.  via  P  =  93  +  -8^  ^  I0-  =322.2. 


it  «  « 


"  L=  227.3  + 


8.8 
322.0. 


THE    TRANSITION-CURVE    FIELD-BOOK. 


43 


Run  Transition  Curve  of  10  chords  of  22.73  eacJl 
from  S  to  Z;  then  the  8°  48'  curve  for  6  —  a 
r=i8°  20'  —  10°  =  8°  20'  to  P.C.C. 

PROBLEM  VII. 

Given  the  intersection  angle  I  of  two  tangents  connected 
by  the  circular  curve  D°,  to  change  the  degree  of 
curve  without  moving  the  position  of  its  middle  point, 
and  insert  Transition  Curves  between  tangents  and 
circular  curve. 


44  TME    TRANSITION-CURVE    FIELD-I3OOK. 

From  the  figure,  M' B  —  MP  —  BP'.  But 
M'B  =  R'  versin  |7,  J/7>  —  R  versin  £7,  and 
.#/>'  =/;  hence 

R'  versin  \I  =  R  versin  \I  —  p.      .     (35) 
>     Assume/",  and 

R'«  =  R-  P. .  (36) 

versin  \I 

Find  the  value  of  Z>'°  from  radius  R'a  to  the 
nearest  even  minute,  and  take  the  exact  radius  R' 
corresponding  to  Z>'°.  Then  from  (35), 

/  =  (R  -  7?')versin  |7.      .     .     .     (37) 

x  =  LK  =  AB+  BP'  =  R'  versin  a  +/, 
and 

q  =  Fx  —  F(R'  versin  a+p).   .     (38) 

IS  =  IP9  -  P'K  +  KS 

=  (*'+/)  tang  i/-^o  +  ^-  .     .      '   .     (39) 

Rule.  Find  R'  of  D'°  from  (36),  and/  from  (37). 
Take  Q  next  lower  than  /Z>'°,  and  from  the  same 
column  Z,  F,  Y,  C,  and  of. 

Find  ^  from  (38). 

Q          „         Z  « 

7=^  •  #^y"* 

Find  7-5  from  (39). 


THE    TRANSITION-CURVE    FIELD-BOOK.  45 

EXAMPLE.  /=  40,    D°  =  6. 

Assume    pa  —  4,     then     R'a  —  955.4  --  -  — 

.060308 

—  955-4  "~  66.3  —  889.1.  To  the  nearest  even  min- 
ute Z>'°  =  6°  26'  and  R'  =  891.1. 

P  =  (955-4  -  89r.i).o6o3o8  =  3.878; 

3.878  X  6°.43  =  24.9.  Next  lower  value  of  Q  is 
Q9  —  19.08;  and  F9  =  .25028,  a9  =  8°  06',  Z9  =  1620, 
C.  =  807.3,  Y9=  898.22. 

0.25028  (891.1  X  0.009976  +  3.878)  =  q  —  3.1955; 


1620 


271.36 
=  n  =  271.36;  -      *—  =  ^  —  30.11;. 


5-97  9 

ro          to  \         £.  8°7-3     ,     30-I5 

is  =  (89i.i  +  3-9)0.36397  -  -^  +  ' 

=  325.7  -  125.5  +  270.8  =  471.0. 

At  L  the  Transition  Curve  with  rate  of  curva- 
ture D"°  =  5°  59'  connects  with  circular  curve 
D'°  =  6°  26'. 

EXAMPLE.  Same  as  last,  except  that  the  value 
of  Q  taken  is  nearest  top  X  D'°. 

^'—891.1  and  /  =  3.878  are  found  as  before. 
The  nearest  Q  to  24.9  is  Q10  —•  29.081;  and 
F^  =  0.25047,  a10  =  10°,  Z19  =  2000,  C10  =  994-9, 
^,o  =  996  -97- 


46  THE    TRANSITION-CURVE    FIELD-BOOK. 

.25047  (891.1  X  0.015192  +  3-878)  —q  — 


^°      =  D""  =  6°  .66  7-6°  40'; 
4.362 


OO  3OO 

=  "  =   °°;='= 


=  ,8,,,  +  3.9)  0.363,7  - 


=  325-7  —  !54-6  +  299.1  =  470.2, 

At  L  the  Transition  Curve  with  rate  of  curva- 
ture Z>"°  =  6°  40'  connects  with  circular  curve 
D'°  =  6°  26'. 


PROBLEM  VIII. 

To  pass  a  circular  curve  through  a  point  H  (Fig.  6) 
given  by  its  radial  distance  HI  from  I,  //fo  circular 
curve  to  be  connected  by  Transition  Curves  with  tan- 
gents intersecting  at  I. 

Rule.  Find  R,  the  radius  of   the   circular   curve 
touching    both    tangents,  by    the    formula    R  — 

TT  T 

,  and  proceed  as  in  Problem  VII. 

ex  sec  !/ 

EXAMPLE.  /  =  40,     HI  =  61.32,    R  =  ~^r^g  — 

61.32  X. 93960  ,    ^0       ,0      ~u 

— J yj?  y  —  955.4,  and  D    —  6  .     The  remain- 

.06031 

<ier  of  this  example  is  exactly  as  in  the  examples 
given  under  Problem  VII,  if  pa  be  assumed  =  4.0. 


THE    TRANSITION-CURVE    FIELD-BOOK.  47 

EXAMPLE  2,  Problem  VIII. 

Given  7=6o,  #7=125,  -£=—-=857.9. 

•1547 


Assume/*  —  5,  R'a  ==  857.9 —  820.5. 

*     OO s i 0 


J = 

13975 
Take  R'  —  819.0,  Z>'°  =  7°;  then  (Form  a3 

P=  (857-9-819.0)  .133975  =  5-212. 


5212  X  7  —  36.7.  Next  lower  Q  is  (?JO  =  29.081; 
and  Fl(}  =  0.25047,  arlo  =  10°,  Z10  =2000,  6^  =  994.9, 
^,0  =  996-97- 

.25047  (819.0  X  0.01592  +  5.212)  =  q  —  4-571; 


^^1  =  jy>°  =  6o    62  = 
4.571 

2OOO  ^14.4 

' 


/^  =  (8I9.o  +  5.2)  .57735  -         ?  +  52 

=    475.8   -     I42.I    +    313.5    =    647.2. 

At  L  the  Transition  Curve  rate  is  6°  22',  the 
circular  curve  7°. 

PROBLEM  IX. 

With  track  laid  over  a  circular  curve  of  small  central 
angle  connecting  two  tangents,  to  shift  the  middle  of 
curve  outward  and  ends  of  curve  inward  and  connect 
with  same  tangents  by  Transition  Curves. 

In  order  to  hold  the  road-bed  as  nearly  as  may 
be,  if  the  shift  at  the  middle  is  h  outward,  that  at 
the  ends  should  be  about  2/1  inward. 


48  THE    TRANSITION-CURVE    FIELD-BOOK. 

In  the  figure  (7),  OP  -  R  of  Z>°,  and  O' B' 
of  D'\ 


From  the  figure,  M* B'  —  MB  —  AB. 
But  M'B'^R'  versin  -|7,  MB=(R+K)  versin  •$/,. 
and  ^4-Z?  =/  -(-  /^;  hence 

7?'  versin  \I  •=  (R  +  ti)  versin  |7—  (/  +  X).  .     (40) 
Assume/a=:  2/2;  then 


versin  \I 


THE    TRANSITION-CURVE    FIELD-BOOK.  49 

Find  the  value  of  D'°  from  radius  R'a  to  the 
nearest  even  minute,  and  take  the  exact  radius  R' 
corresponding  to  Z>'°.  Then,  from  (40), 

/=(.#  +  //  —  R')  versin  \I  —  h.     .     .     (42) 
x  =  LK  —  GB'  +  AP  =  R'  versin  a  +/; 
q  —  Fx  =  F(R'  versin  a  +/).    .     .     .     (43) 
SP  =  SK  -  KP'  +  P'K'  -  K'P\ 

SK=  —  ',  KP'  =  R'  sin  a  =  ^5 

100  // 

/"A"'  =  ^'  sin  |7;  K'P  =  (^  +  //)  sin  |7. 
Therefore 

^  -  f^o  -  ^o  -  (^  +  h  -  R'}  sin  |7.    (44) 

J?///<?.   Find  7?'  of  D'°  from  (41),  and  p  from  (42). 
Take  (2  next  lower  than  /Z>'°,  and  from  the  same 
column  X,  F,  Yy  C,  and  a. 
Find  q  from  (43). 

<2  _     ry,*  Z          _„        n     __ 

--  >     ^775-    »,     Tvr- 

Find  5P  from  (44). 

EXAMPLE.  D°  =  5°,  7—  30°,  h  --  2. 
Radius  of  D°  =  1146.3,  and 

1146.3  4-  2  — =  ^?/a  =  072. 2. 

.034074 

Take  R'=  971.5,  corresponding  to  Z>'°  =  5°  54'= 
5°. 9.     Then  (1146.3  4-2  —  971.5)  .034074  —  2  =/  = 


50  THE    TRANSITION-CURVE    FIELD-BOOK. 

4.024;  pD'°  —  4.024  X  5.9  =  23.24.  From  column 
^9>  (?9  =  19.08;  and  Z9  =  1620,  C9  =  807.3,  F9  = 
898.22,  JF9  =  .25028. 

.25028  (971.5  X  .00998  +  4.024)  =  q  —  3.434; 
-^— -  =  ^"°  =  5°-556  =  5°  33?'; 

_       .    _     .  J       'JJ  J          «JO«J       7 


1620  201.6 

__  =  ,  =  29,6;___  =  ,  = 


==.  291.0  —    136.8  —  45-8  =   108.4. 


From  ^  to  middle  of  curve  £>°  =  SP  +  ^4,-°°= 


108.4  H ~  :=:  108.4  H~  3°°  ==  408.4.      From    61 

to  middle  of  curved'0  =  n  +  ^ — -=—^=291.6+ 

-  =  291.6  +  116.9  —  408.5.     The    mid- 
o  *  v 

die  of  D'°  being  outside  of  D°  makes  the  new  line 
o.i  the  longer. 

In  this  example  the  rate  of  curvature  of  the 
Transition  Curve  at  L  (sta.  9)  is  5°  33^',  connecting 
there  with  a  5°  54'  circular  curve. 


THE    TRANSITION-CURVE    FIELD-BOOK.  51 


PROBLEM  X. 

With  track  laid  over  two  circular  curves  turning  in 
opposite  directions  and  a  tangent  connecting  them,  to 
change  the  direction  of  the  tangent  and  insert  Transi- 
tion Curves  between  the  circular  curves  and  the  new 
tangent. 

Given  the  degrees  of  the  two  curves  D°  and  D'° 
the  positions  of  the  P.  T.  of  Z>°  and  P.C.  of  Z>'°, 
and  the  length  of  tangent  from  P.  T.  to  P.C. 

Rule.  Assume  approximately  the  gaps  qa  and  q'a 
between  the  circular  curves  and  the  new  position 
of  tangent.  Take  from  table  Q  and  Q'  nearest  to 
qaD^  and  q'aDfc ',  and  from  same  columns  Z,  C,  and 
F,  andZ',  C',  and  Y' . 

Z        Z' 

If  tangent  length  T  is  less  than  about  — -7^54—^75, 

2.D          2JJ 

adopt  a  lesser  value  of  qa  or  q'a  or  of  both,  and 
proceed  as  before.  Then  for  Transition  Curve 
connecting  with  Z>°, 


and  for  Transition  Curve  connecting  with  Z>'°, 


To  find  the  positions  of  Z,  S,  S',  Z',  and  the  dis- 
tance from  ^  to  *$*',  use  the  formulas  following 
Fig.  8. 

In  Fig.  8,  R  and  Rr  are  the  radii  of  the  given 
circular  curves  D°  and  JD'°  connected  by  tangent 
T  from  P.  T  of  D°  to  f,C,  of  D'°, 


52  THE    TRANSITION-CURVE    FIELD-BOOK. 

From    centre    O    with    radius    R  -f-  ^  and  from 
centre    0'    with    radius    R'  -\- q' ^  draw  arcs  as   in 


figure  and  connect  them  by  tangent  PP'.  Prolong 
radius  R'  at  P.C.  and  intersect  it  at  E  by  OE  par- 
allel to  tangent  7T;  and  prolong  radius  R9  +  ^'  at 
/^  and  intersect  it  at  G  by  OG  parallel  to  new 
tangent  PP* '. 


THE    TRANSITION-CURVE    FIELD-BOOK.  53 


__ 
sin  00'  E      sin  OO'E 


OO'  X  sin  OO'G  —  OG  =  PP'. 

Designate  the  angle  OO'E  —  OO'G  by  ft. 

Then  a  —  ft  =  angle  from  P.T.  back  to  Z,  and 
a'  —  /3  =  angle  from  P.C.  forward  to  Z'. 

To  find  K  and  K'  on  tangent  PPf  prolonged,  set 
the  instrument  over  Z,  turn  90  —  a  outward  from 

tangent  at  Z,  and  measure  on  this  course  LK—  — . 

100 

Similarly  at  Z'  turn  90  —  a'  and  measure  L' K'  — 
cX' 

IOO* 

cY 

From  K  measure  KS  =  —  on  tangent  KK ',  and 

100 

put  in  point  S\  similarly  from  Kf  measure  K' S'  = 

c'Y' 

and  put  in  point  S'. 

IOO 

SS'  =  PP9  +  XP  +  P'K'  -  SK  -  S'K' 

r  ~no  T"  ~7v"o       ~~  ~-  •     •     •     (45) 


From  Z  to  L'  via  old  line  =  —  ^"          + 
IPO  (a9  -  /?) 

jyo 

From  Z  to  Z'  via  new  line  =  n  +  6*5'  -f- ;/. 


54  THE    TRANSITION-CURVE    FIELD-BOOK. 

EXAMPLE.  Given  a  10°  curve  to  the  right  and  a. 
5°  curve  to  the  left  with  a  tangent  500  feet  long 
between  them,  to  change  the  tangent  so  that  its 
direction  will  pass  about  6  feet  outside  of  the  10° 
curve  and  about  4  feet  outside  of  the  5°  curve, 
both  circular  curves  to  be  connected  by  Transition 
Curves  with  the  new  tangent. 

For  the  10°  curve  J?  =  573.7,  qa  =  6,  qaD°  = 
6  x  10  =  60.  From  column  -A^12,  QIZ  =  60.302;  and 
C12=  1424.9,  F19=  1192.5. 


60.302  2880 

-  -  —  —  q  —  6.0302:     -  ==  n  =  288.0: 


10  10 

288.0 

-  =  c  =  24. 


For  the  5°  curve  R'  =  1146.3,  q'a  —  4,  'q'«  Z>'°  = 
4  x  5  =  20.      From  column  NQ  ,    Q9  —  19.08;    and 
=  807.3,    y.  =  898.2. 

.  08  1620 

=  n'  =  324.0; 


D  0 


9   • 
-g,  =  tang  OO'E  = 

=  tang  16°  12^'; 
T 


sin  Uu  ±L       .279131 


=  COS  15 


THE    TRANSITION-CURVE    FIELD-BOOK.  55 

OO'X  sin  OO'G=-  1791.2  X  0.25952  =  464.8  =  PP'; 
OO'E  -  OO'G  =  16°  12  J'  -  15°  o2-i'  =  1°  10'  =  ft. 

a.  —  ft  =  14°  24'  —  i°  10'  =  13°  14'  =  angle    from 
P.  T.  back  to  L. 

a'  -  ft  =    8°  06'  -  i°  10'  =    6°  56'  =  angle     for- 
ward from  P.C.  to  L9. 

Fix  the  point  L  by  going  back  from  P. T.  for 
13°  14'  of  arc,  or  132.3  feet.  Set  instrument  over  L, 
turn  90°  —  a  —  90  —  14°  24'  =  75°  36'  outward  from 

cX 

tangent  at  L.  and  on   this  course  measure  -  -  = 

100 

24  X  100.083 

—  =  24.02  to  point  K. 
100 

Similarly  for  L9  turn  81°  54'  outward  from  tan- 

r'Yf 

gent  and  measure  -  -   —  =  15.25  to  K9 . 

cY 

From  K  measure  —  =  286.2  on  KK*  to  S\  and 
100 

c'Y' 
from  K'  measure  -    -  =  323.4  on  K9 K  to  S'. 

TOO 

C          r"         rV         r'V 

ocr  __  ppr  _i    _T_     i    _T_ 

—  •"    7~)°      "^    7~)'°  ~" 

=  464.8  +  142.5  -f  161.5  —323-4— 286.2  =  159.2. 

Except  as  a  test  of  the  correctness  of  the  rest  of 
the  work  it  is  not  necessary  to  calculate  or  meas- 
ure SS'. 

From  L  to  L9  via  old  line  =  IO°Xl3-233  +  5QQ  + 

10 

.00x6.933 

100 


56  THE    TRANSITION-CURVE    FIELD-BOOK. 

From  L  to  L'  via  new  line  =  288.0  -f-  159.2  -f 
324.0  =  771.2. 

The  line  from  L  to  L'  may  be  run  in  contin- 
uously by  angles,  or  the  two  Transition  Curves 
may  be  laid  off  from  S  to  L  and  from  Sf  to  Z' 
either  by  angles  or  ordinates. 


PROBLEM  XL 

When  a  Transition  Curve  is  run  in  and  does  not  prop- 
erly connect  at  L  or  at  S,  as  the  case  may  be,  to  make 
the  proper  adjustment  and  location.  (See  Fig.  i.) 

Rule.  Set  the  instrument  over  a  trial  L  on  the 
circular  curve,  turn  into  tangent,  and  intersect  the 
given  tangent  at  T.  Move  to  T  and  measure  the 
intersection  angle  LTK.  If  L TK  =  a,  the  trial 
point  set  over  is  correct  for  L,  and  the  distance  LT 
multiplied  by  sin  a  and  divided  by  the  correspond- 
ing value  of  X  from  the  table  gives  the  chord 
length  c  with  which  the  Transition  Curve  from  L 
to  S  is  to  be  rerun. 

If  the  intersection  angle  LTK  as  measured  is 
not  equal  to  a,  make  it  equal  by  moving  the  trial 
L  backward  or  forward  on  the  circular  curve 
through  a  central  angle  equal  to  the  difference 
between  the  angles  LTK  and  a.  When  L  is  cor- 
rectly located,  set  over  it  and  turn  into  tangent. 
From  this  tangent  turn  off  outwardly  the  angle 
90°  —  a,  and  on  this  course  (perpendicular  to  SK} 
intersect  the  given  tangent  in  K  and  measure  LK. 
Then 

:  TOO  X~      cY 

=  f,     ana 


THE    TRANSITION-CURVE    FIELD-BOOK.  57 

Be  careful  to  take  table  quantities  from  the  col- 
umn giving  the  proper  value  of  a. 

Super-elevation  of  outer  rail  on  curves. 

The  difference  of  elevation  e  between  the  inner 
and  outer  rails  on  Transition  Curves  should  in- 
crease regularly  from  °  at  -S1  to  its  maximum  at  L. 
This  difference  of  elevation  for  a  4'  8y  gauge 
(and  proportionally  for  other  gauge  widths)  is 
closely  approximated  for  all  velocities  and  rates  of 
-curvature  by  taking  ^  =  .000056  V*D°,  in  which 
V  is  the  velocity  in  miles  per  hour  and  DQ  the 
degree  of  curve. 

For  example:  on  a  Transition  Curve  of  9  stations 
connecting  with  a  6°  curve,  the  superelevation  e 
of  the  outer  rail  at  L  and  around  the  6°  curve, 
adjusted  for  40  miles  an  hour,  will  be 

e  =  .000056  X  4°9  X  6  =  0.5376; 

and   for   stations  i,  2,  3,  4,  etc.,  of    the    Transition 
Curve  the  differences  in  elevation  are  0.5376  X  -J-, 
X  f   X  |,   X  |,  etc. 
In    surfacing   track    the    inner   rail    should    be 

lowered  — ,  and  the  outer  rail  raised  — ;  and  this  is 

2  2 

conveniently  done  by  setting  stakes  at  a  distance 
equal  to  the  width  of  the  gauge  plus  the  width  of 
the  rail-head  on  each  side  of  the  centre  stake  with 
the  top  of  the  latter  at  its  proper  elevation,  and 
the  tops  of  the  inner  and  outer  stakes  respectively 
e  below  and  e-  above  it 


TABLES. 


RADII  OF  DEGREES  OF  CURVE. 


/ 

0° 

1° 

2° 

£° 

40                  50 

/ 

Hadius. 

Radius. 

Radius. 

Radius. 

Radius.      Radius. 

(I 

Infinite" 

5729  65 

2864  .  93 

1910  08 

1432.69 

1146  28 

0 

1 

343;  75 

5635.72 

2841.26 

1899.53 

1426.74 

1142.47 

1 

2 

171887 

5544  83 

2817.97 

1889.09 

1420.85 

1138.69 

2 

y 

1  1  4592 

5456.82 

2795.06 

1878.77 

1415.01 

1134.94 

3 

4 

85913.7 

5371.56 

2772.53 

1868.56 

1409.21 

1131.21 

4 

5 

68754.9 

5288  .  92 

2750.35 

1858.47 

1403.46 

1127.50 

5 

6 

57295.8 

5208.79 

2728.52 

1848.48 

1397.76 

1123.82 

6 

"j1 

49110.7 

5131.05 

2707  04 

1838.59 

1392.10 

1120.16 

7 

8 

42971.8 

5055.59 

2685.89 

1828  82 

1386.49 

1116.52 

8 

9 

3S197.2 

49.S2  33 

2665.08 

1819.14 

1380.92 

1112.91 

9 

10        34377.5 

4911.15 

2644.58 

1809.57 

1375.40 

1109.33 

10 

11        31-252.3 

4S41.98 

2624.39 

1800.10 

1369.92 

1105.76 

11 

12  !     28617.8 

4774  .  74 

2604  51 

1790.73 

1364.49 

1102.22 

12 

13 

26444.2 

4709.33 

2584  .  93 

1781.45 

1359.10 

1098.70 

13 

14 

24555.4 

4645.69 

2565  .  65 

1772.27 

1353.75 

1095.20 

14 

15       22918.3 

4583.75 

2546.64 

1763.18 

1348.45 

1091.73 

15 

1»> 

21485.9 

4523.44 

2527.92 

1754.19 

1343.18 

1068.28 

16 

17 

20222.1 

4464.70 

2509.47 

1745.29 

1337  96 

1084.85 

17 

18 

19098  6 

4407.46 

2491.29 

1736.48 

1332.77 

1081.44 

18 

in 

18093  4 

4351  67 

2473.37 

1727.75 

1327.63 

1078  05 

19 

20        17188.8 

4297.28 

2455  .  70 

1719.12 

1322.53 

1074.68 

20 

21 

16370.2 

4244.23 

2438.29 

1710.56 

1317.46 

1071.34 

21 

22 

15626.1 

4192.47 

2421.12 

1702  10 

1312.43 

1068.01 

22 

2-5 

14946  7 

4141  96 

2104.19 

1693.72 

1307.45 

1064.71 

23 

21 

14323.6 

4092  66 

2387  50 

1685.42 

1302.50 

1061.43 

24 

25 

13751.0 

4041.51 

2371  04 

1677  20 

1297.58 

1058.16 

25 

26 

J3*».l 

3997.4'.) 

2354  80 

1669.06 

1292  71 

1054.92 

26 

27 

12732.4 

3951.54 

2338.78 

1661.00 

1287.87 

1051.70 

27 

28 

12277.7 

3906.  64 

232-2.98 

1653.01 

1283.07 

1048.48 

28 

29 

11854.3 

3S62.71 

2307.39 

KJ45.11 

1278.30 

1045.31 

29 

30 

11459.2' 

3819.83 

2292.01 

1637.28 

1273.57 

1012.14 

30 

31 

11089  6 

3777.85 

2276.84 

1629.52 

1268.87 

1039.00 

31 

32 

10743.0 

3736  7i) 

2261.86 

1621.84 

1264.21 

1035.87 

32 

33 

10417.5 

3696.61 

2247.08 

1614.22 

1259.58 

1032.76 

33 

31 

10111.1 

3657.29 

2232.49 

1606.68 

1254.98 

1029.67 

34 

35 

9822.18 

36  16.  SO 

221  S.  09 

1599.21 

1250.42 

1026.60 

35 

36 

9549.  3  t 

3581    10 

2203.87 

1591.81 

1245.89 

1023.55 

36 

37 

9291.29 

3544.19 

2189.84 

1584.48 

1241.40 

1020.51 

37 

8H 

9046.75 

3508  0-3 

2175.98 

1577.21 

1236.94 

1017.49 

38 

39 

8814.78 

3472  59 

2162.30 

1570.01 

1232.51 

1014.50 

39 

40 

8594.  4  i 

3  437.  £7 

2148.79 

1562.88 

1228.11 

1011.51 

40 

41 

8384.80 

3403.83 

2135.44 

1555.81 

1223.74 

1008.55 

41 

42 

8185.16 

3370.46 

2122.26 

1548.80 

1219.40 

1005.60 

42 

43 

7994.81 

3387.74 

2109  24 

1541.86 

1215.09 

1002.67 

43 

41 

7813.11 

3305.  «5 

2096.39 

1534.98  , 

1210.82 

999.76 

44 

45 

7639.49 

3274.17 

2083.68 

1528.16 

1206.57 

996.87 

45 

46 

7473.42 

3243.29 

2071.13 

1521.40 

1202.36 

993.99 

46 

47 

7*14.41 

3-212.98 

2058.73 

1514.70 

1198.17 

991.13 

47 

4S 

7162.03 

3183.23 

2046.48 

1508.06 

1194.01 

988.28 

48 

4'.) 

7015.87 

3151.03 

2034.37 

1501.48 

1189.88 

985.45 

49 

50 

6875.55 

3125.  3J 

2022.41 

1494.95 

1185.78 

982.64 

50 

M 

6740  74 

3097.20 

2010.59 

1488.48 

1181.71 

979.84 

51 

6611.12 

3069.55 

1998.90 

1482.07 

H77.C6 

977.06 

52 

M 

6486.  3S 

3042.39 

1987.35 

1475.71 

1173.65 

974.29 

53 

54 

6366.26 

3015.71 

1975.93 

1469.41 

1169.66 

971.54 

54 

55 

6250.51 

2989.48 

1964.64 

1463.16 

1165.70 

968.81 

55 

50 

6138.90 

2963.71 

1953.48 

1456.96 

1161.76 

966.09 

56 

57 

6031  .  20 

2938.39 

1942.44 

1450.81 

1157  85 

963.39 

57 

58 

5927.22 

2913.49 

1931.53 

1444.72 

1153.97 

960.70 

58 

59 

5826.76 

28H9.01 

1920.75 

1438.68 

1150.11 

958.02 

59 

60 

5729.65 

28G1.93 

1910.08 

1432  (50 

1146.28 

955.37 

60 

RADII  OF  DEGREES  OF  CURVE. 


/ 

6° 

7°   !   8° 

9° 

1O° 

11° 

t 

Radius. 

Radius.  !  Radius. 

Radius. 

Radius. 

Radius. 

~F 

955.37 

819.02 

716.78 

637.28 

573.69 

521  .  67 

~TT 

i 

952.72 

817.08 

715.29 

636.10 

572.73 

520.89 

i 

2 

950.09 

815.14 

713.81 

634.93 

571.78 

520.10 

2 

3 

947.48 

813.22 

712.33 

633.^6 

570.84 

519.32 

3 

4 

944.88 

811.30 

710.87 

632.  <>0 

509.90 

518.54 

4 

5 

942.29 

809.40 

709  40 

631.44 

568.96 

517.76 

5 

6 

939.72 

807.50 

707.95 

630.29 

568.02 

516.99 

6 

7 

937.16 

805.61 

706.49 

629.14 

567.09 

516.21 

7 

8 

934.62 

803.73 

705.05 

0*7.99 

566.16 

515.44 

8 

9 

932.09 

801.86 

703.61 

626.85 

565.23 

514.68 

9 

10 

929.57 

800.00 

702.18 

625.71 

564.30 

513.91 

10 

11 

927.07 

798.14 

700.75 

624.58 

563.38 

513.15 

11 

12 

924.58 

796.30 

699.33 

623.45 

562.47 

512.38 

1* 

13 

922.10 

794.46 

697.91 

622.33 

561.55 

511.63 

13 

14 

919.64 

792.63 

696  50 

621.20 

560.64 

510.87 

14 

15 

917.19 

790.81 

695.09 

620.09 

559.73 

510.12 

15 

16 

914.75 

789.00 

693.70 

618.97 

558.82 

509.36 

16 

17 

912.33 

787.96 

692.30 

617.87 

557.92 

508.61 

17 

18 

909.  02 

785  40 

6',)0.91 

616.76 

557.02 

507.87 

18 

19 

907.52 

783.62 

689.53 

615.66 

556.12 

507.12 

19 

20 

905.13 

781.84 

688.16 

614.56 

555.23 

506.38 

20 

21 

902.76 

780.07 

686.78 

613.47 

554.34 

505.63 

21 

2:2 

900.40 

778.31 

685.42 

612.38 

553.45 

504.90 

22 

23 

898.05 

776  55 

684.06 

611.30 

552.56 

504.16 

23 

24 

895.71 

774.81 

682.70 

610.21 

551.68 

503.42 

24 

25 

893.39 

773.07 

681.35 

609.14 

550.80 

502.69 

25 

26 

891.08 

771.34 

680.01 

608.06 

549  92 

501  .  96 

26 

27 

888.78 

769.61 

678.67 

606.99 

549.05 

501.23 

27 

28 

886.49 

767.90 

677.34 

605.93 

548.17 

500.51 

28 

29 

884.21 

766.19 

676.01 

604.86 

547.31 

499.78 

29 

30 

881.95 

764.49 

674.69 

603.81 

546.44 

499.06 

30 

31 

879.69 

762.80 

673.37 

602.75 

545.58 

498.34 

31 

32 

877.45 

761.11 

672.06 

601.70 

544.71 

497.62 

32 

33 

875.22 

759.43 

670.75 

600.65 

543.86 

496.91 

33 

34 

873.00 

757.76 

669.45 

599.61 

543.00 

496.19 

34 

35 

870.79 

756.10 

668.15 

598.57 

542.15 

495.48 

35 

36 

868  60 

754.45 

666.86 

597.53 

541.30 

494.77 

36 

37 

866.41 

752.80 

665.57 

596.50 

540.45 

494.07 

37 

38 

864.24 

751.16 

664.29 

595.47 

539.61 

493.36 

38 

39 

862.07 

749  52 

663.01 

594.44 

538.76 

492.66 

39 

40 

859.92 

747.89 

661.74 

593  42 

537.92 

491.96 

40 

41 

857.78 

746.27 

660.47 

592.40 

537.09 

491.26 

41 

42 

855.65 

744.66 

659.21 

591.38 

536.25 

490.56 

42 

43 

853.53 

743.06 

657.95 

590.37 

535.42 

489  86 

43 

44 

851.42 

741.4*6 

656.69 

589.36 

534.59 

489.17 

44 

45 

849.32 

739.86 

655.45 

588.36 

533.77 

488.48 

45 

46 

847.23 

738.28 

654.20 

587.36 

532.94 

487.79 

46 

47 

845.15 

736.70 

652.96 

586.36 

532.12 

487.10 

47 

48 

843.08 

735.13 

651.73 

585.36 

531.30 

486.42 

48 

49 

841.02 

733.56 

650.50 

584.37 

530.49 

485.73 

49 

50 

838.97 

732.01 

649.27 

583.39 

529.67 

485.05 

50 

51 

836.93 

730.45 

648.05 

582.40 

528  86 

484.37 

51 

52 

834.90 

728.91 

646.84 

581.42 

528.05 

483.69 

52 

53 

832.89 

727.37 

645.63 

580.44 

527.25 

483.02 

53 

54 

830  88 

725.84 

644.42 

579.47 

526.44 

482.34 

54 

55 

828.88 

724.31 

643.22 

578.49 

525.64 

481.67 

55 

56 

826.89 

722.79 

642.02 

577  53 

524.84 

481.00 

56 

57 

824  91 

721.28 

640.83 

576.56 

524.05 

480.33 

57 

58 

822.93 

719.77 

639.64 

575.60 

523.25 

479.67 

58 

59 

820.97 

718.27 

638.46 

574.64 

522.46 

479.00 

59 

60 

819.02  - 

716.78 

637.28  |  573.69 

521.67 

478.34 

60 

RADII  OF  DEGREES  OF  CURVE. 


r 

12° 

13° 

14° 

15° 

10° 

17° 

f 

Radius. 

Radius. 

Radius. 

Radius. 

Radius. 

Radius. 

~0~ 

478.34 

441.68 

410  28 

383.06 

359.27 

338.27 

~cT 

1 

477.68 

4*1.12 

409  79 

382.64 

358.89 

337.94 

i 

2 

477.02 

443  56 

409.31 

382.22 

3.'8.52 

337.62 

2 

3 

476.36 

410.00 

408.82 

381.80 

358.15 

337.29 

3 

4 

475.71 

430.44 

408  -34 

381.38 

357.78 

336.96 

4 

5 

475.05 

438.88 

407  .  86 

380.96 

357.42 

336.04 

5 

6 

471.40 

433.33 

407.38 

380.54 

357.05 

336  31 

6 

7 

473.75 

437.77 

406.90 

380  13 

356.68 

335  .  1)9 

7 

8 

473.10 

43?.  22 

406.42 

379  71 

356.31 

335.66 

8 

9 

472.46 

436  67 

405.95 

379.29 

355.05 

335.34 

9 

10 

471.81 

436.12 

405.47 

378.88 

355.59 

335.01 

10 

11 

471.17 

435.57 

405.00 

378.47 

355  .23 

334.69 

11 

14 

470.53 

435.02 

404.53 

378.05 

354  >6 

334.37 

12 

13 

460  89 

434.47 

404.05 

377.64 

854.50 

334.05 

13 

14 

469.25 

433.93 

403.  53 

377.23 

354.13 

333.73 

14 

15 

468.61 

433.39 

403.11 

3  IT,.  82 

353.77 

333.41 

15 

16 

467.98 

432.84 

402.65 

3J6.41 

353.41 

333.09 

16 

17 

467.35 

432.30 

402.18 

376.(»0 

353.05 

332.77 

17 

13 

466.72 

431.76 

401.71 

375.60 

352.70 

332.45 

18 

19 

466.09 

431.23 

401.25 

275.19 

352.34 

332.13 

19 

20 

465.46 

430.69 

400.78 

37'4.79 

351.98 

331.82 

20 

21 

461.  as 

430.15 

400.32 

374.38 

351.62 

331.50 

21 

22 

464.21 

429.62 

399.86 

373.98 

S51.27 

331.18 

22 

23 

463.59 

429.09 

399.40 

373.57 

350.91 

330.87 

23 

24 

462.97 

428.56 

398.94 

373.17 

350.56 

330.56 

24 

25 

462  35 

428  03 

398.48 

372.77 

350.21 

330.24 

25 

26 

461.73 

427.50 

39802 

372.37 

319.85 

329.93 

26 

27 

461.11 

426.97 

397.56 

371.97 

349.50 

329.62 

27 

28 

460.50 

426.45 

397.11 

371.57 

349.15 

329.30 

28 

29 

459  89 

425.92 

396.65 

371.18 

348.80 

328.99 

29 

30 

459.28 

425.40 

396.20 

370.78 

348.45 

328.68 

30 

31 

458.67 

424.87 

395.75 

370.38 

348.10 

328.37 

31 

32 

458  06 

424.35 

395.30 

369.99 

347.75 

328.06 

32 

«33 

457.45 

423.83 

394.85 

369  60 

347.40 

327.75 

33 

34 

456.85 

423.32 

394.40 

369.20 

347.06 

327.44 

34 

35 

456.25 

422.80 

393.95 

368.81 

346.71 

327.14 

35 

36 

455.65 

422.28 

393.50 

368.42 

346.37 

326.83 

36 

37 

455.05 

421.77 

393.05 

368.03 

346.02 

326.52 

37 

38 

454.45 

421.26 

392.61 

367.64 

345.68 

326.22 

38 

39 

453.85 

420.74 

392.16 

367.25 

345.33 

325.91 

39 

40 

453.26 

420.23 

391.72 

366.86 

344.99 

325.60 

40 

41 

452.66 

419.72 

391.28 

366.47 

344.65 

325.30 

41 

42 

452.07 

419.22 

390.84 

366.09 

344.31 

325.00 

42 

43 

451.48 

418.71 

390.40 

365.69 

343.97 

324.70 

43 

44 

450.89 

418.20 

389.96 

365.31 

343.63 

324  39 

44 

45 

450.31 

417.70 

389.52 

364.93 

343.29 

324.09 

45 

46 

449  72 

417.20 

389.08 

364.55 

342.95 

323.79 

46 

47 

449.14 

416.69 

388.65 

364.16 

342.61 

323.49 

47 

48 

448.56 

416.19 

388.21 

363.78 

342.27 

323.18 

48 

49 

447.97 

415.69 

387.78 

363.40 

341.93 

322.89 

49 

50 

447.39 

415.19 

387.34 

363.02 

341.60 

322.59 

50 

51 

446.82 

414.70 

386.91 

362.64 

341.26 

322.29 

51 

52 

446.24 

414.20 

386.48 

362.26 

340.93 

321.99 

52 

53 

445.67 

413.71 

386.05 

361.89 

340.59 

321.69 

53 

54 

445.09 

413.21 

385.62 

361.51 

340.26 

321.39 

54 

55 

444.52 

412.72 

385.19 

361.13 

339.93 

321.10 

55 

56 

443  95 

412.23 

384.77 

360.76 

339.59 

320.80 

56 

57 

443.38 

411.74 

384.34 

360.38 

339.26 

320.51 

57 

58 

442.81 

411.25 

383.91 

360.01 

338.93 

320.21 

58 

59 

442.25 

410.76 

383.49 

359.64 

338.60 

319.92 

59 

60 

441.68 

410.  2S 

383.06 

359.27 

338.27 

319.62 

60 

RADII  OF  DEGREES  OF  CURVE. 


f 

18° 

19° 

20° 

21° 

33° 

23° 

/ 

Radius. 

Radius. 

Radius,  l  Radius 

Ktidius.  ;  Fad  us. 

0 

319.62 

302.94 

287.  '.14 

274.37 

•J02  04     !     250  79 

~T 

1 

319  33 

302.68 

287.70 

274  1C) 

261.85     !     250.61 

i 

2 

319  04 

302.42 

287.40 

273  94 

261.05          250.  4  d 

2 

a 

318.74 

302.10 

287.23 

273.73 

201.45         25026 

3 

4 

318.45 

801.89 

x86  99 

273  51 

201  20         250.08 

4 

5 

318.16 

301.  (53 

286  76 

273.30 

261.06         249.90 

5 

6 

317.87 

301.37 

2«6.52 

373.08 

360  87 

249  72 

0 

7 

317.58 

301.11 

286.29 

272.  H7 

260.68 

249  54 

7 

8 

317.29 

300  85 

286.05 

272.66 

260  48 

249  87 

8 

9 

317.00 

300.59 

285.82 

272  45 

200  29 

249  19 

9 

10 

316.71 

300.33 

285.58 

272.23 

260.10         2-19.01 

10 

n 

316.43 

300.07 

285.35 

272.02 

259.90 

249.84 

11 

12 

316  14 

299.82 

285.12 

271.81 

259.71 

248.06 

12 

13 

315.85 

299.56 

284.88 

271.60 

259.52 

248.48 

13 

14 

315.57 

299  30 

284.65 

271.39 

*6».83 

248.31 

14 

15 

315.28 

299.04 

284.42 

271.18 

259.13 

248.13 

15 

16 

315.00 

298.79 

284.20 

270.97 

258  94 

247.96 

16 

17 

314  71 

298.53 

283.96 

270.76 

258.75 

247.78 

17 

18 

314  43 

29S.28 

283.73 

270.55 

258.56 

247.61 

18 

19 

3  4.14 

298.02 

283.50 

270.34 

258.37 

247.43 

19 

20 

313.60 

297.77 

283.27 

270.13 

258.18 

247.20 

20 

21 

313.58 

297.51 

283.04 

269.92 

257  99 

247.08 

21 

22 

313.29 

297  .  20 

282.81 

269  71 

257.80 

246.91 

22 

23 

313  01 

297.01 

282  58 

269.51 

257.61 

246.74 

23 

24 

312.73 

290.  75 

282.35 

269.30 

257.42 

240.50 

24 

25 

312.45 

296.50 

282.12 

269  09 

257.23 

240  39 

25 

20 

312.17 

296  25 

281.89 

268.89 

257  04 

246.22 

26 

27 

311.89 

296  00 

281.67 

268.68 

256.85 

246.04 

27 

28 

311.01 

295  75 

281.44 

268.47 

250.07 

245.87 

28 

29 

311.33 

295  50 

281.21 

268.27 

256.48 

245.70 

29 

30 

311.00 

295.25 

280.99 

268  00 

256.29 

245.53 

30 

31 

310.78 

295.00 

280.76 

267.86 

256  .10 

245.36 

31 

32 

310.50 

294.75 

280.54 

267.6*) 

255.92 

245.19 

32 

33 

310.23 

294.50 

280  31 

207.45 

255.73 

245.02 

33 

34 

309  95 

294  25 

280.09 

207.  24 

255.54 

244.84 

34 

3:> 

309.67 

294.00 

279.86 

207.04 

255.36 

244.07 

35 

36 

309.40 

293.76 

279.64 

266.84 

255.17 

244.50 

36 

37 

309.12 

293.51 

279.42 

260.  6^ 

254  99 

244.33 

37 

38 

308.85 

293  26 

279.19 

260.43 

254.80 

244.16 

38 

39 

308.58 

293  01 

278.97 

206.23 

254  62 

243.99 

39 

40 

308.30 

29->.77 

278.75 

266.02 

254.43 

243.83 

40 

41 

308.03 

292.52 

278.52 

265.82 

254.25 

243  66 

41 

42 

307.76 

292.28 

278.30 

265.62 

254.06 

243.49     !  42 

43 

307.49 

292.03 

278.08 

265.42 

254.88 

243.32 

43 

44 

307.22 

291.79 

277.86 

265.22 

253.70 

243.15 

44 

45 

306  95 

291  .  55 

277.64 

265  02 

2£3.51 

242.98 

45 

46 

306  68 

291.30 

277.42 

264.82 

253.33 

242  U 

46 

47 

306.41 

291.06 

277.20 

264.62 

253  14 

242.64 

47 

48 

306.14 

290.82 

276.98 

264.42 

252.96 

242.48 

48 

49 

305.87 

290.58 

276.76 

264.22 

252.78 

242.31 

49 

50 

305.  6C 

290.33 

276.54 

264.02 

252.60 

242.14 

50 

51 

305.33 

290.09 

276.32 

263.82 

252.42 

241.98 

51 

52 

305  06 

289.85 

276.10 

263.62 

252  24 

241.81 

52 

53 

304.80 

289.61 

275.89 

263.42 

252.05 

241.64 

53 

51 

304  53 

289.37 

275.67 

263.22 

251.87 

241.48 

54 

55 

304.27 

289.13 

275.45 

263.03 

251.69 

241.31 

55 

56 

304.00 

288.89 

275.23 

262.83 

251.51 

241  15 

56 

57 

303.73 

288.65 

275.02 

262.63 

251.33 

240.98 

57 

58 

303.47 

288.41 

274.80 

262.44 

251.15 

240.82 

58 

59 

303.21 

288.18 

274.58 

262.24 

250.97 

240.65 

59 

60 

302.94 

287.94 

274  37 

262.04 

250  79 

240.49 

60 

RADII  OP  DEGREES  OF  CURVE. 


/ 

34° 

25° 

36° 

27° 

38« 

29° 

if 

Radius 

Radius. 

Radius. 

Radius. 

Radius. 

Radius. 

"(T 

240.49 

231.01 

222.27 

214.18 

206.68 

199.70 

"IT 

1 

240.32 

230.86 

222.13 

214.05 

206.56 

199.58 

1 

2 

240.16 

230.71 

221.99 

213.92 

206.44 

199.47 

2 

3 

239  09 

230.56 

221.85 

213.79 

206.32 

199.36 

3 

4 

239.83 

230.41 

221.71 

2i3.ru 

206  20 

199.25 

4 

5 

239.67 

230.26 

221.57 

213.54 

206.08 

199  14 

5 

6 

239.50 

230.11 

221.43 

213.41 

205.96 

199.02 

6 

7 

239  34 

229.96 

221.30 

213.28 

205.84 

198.91 

7 

8 

239.18 

229.81 

221.16 

213.15 

205.72 

198.80 

8 

9 

239.01 

229.66 

221.02 

213.02 

205.60 

198.69 

9 

10 

238.85 

229.51 

220.88 

212.89 

205.48 

198.58 

10 

11 

238.69 

229.36 

220.74 

212.77 

205.36 

198.47 

11 

12 

238.53 

229.21 

220.60 

212.64 

205.24 

198  36 

12 

13 

238.37 

229.06 

220.47 

212.51 

205.12 

198  25 

13 

14 

238.21 

228.91 

220.33 

212.38 

205.00 

198.14 

14 

15 

238.04 

228.76 

220.19 

212.25 

204.89 

198  03 

15 

16 

237.88 

228.61 

220.05 

212.13 

204.77 

197.92 

16 

17 

237.72 

228.46 

219.92 

212.00 

204.65 

197.81 

17 

18 

237.56 

228.32 

219.78 

211.87 

204.53 

197.70 

18 

19 

237.40 

228.17 

219.64 

211.75 

204.41 

197.59 

19 

"20 

237.24 

228.02 

219.51 

211.62 

204.30 

197.48 

20 

21 

237.08 

227.87 

219.37 

211  .49 

204.18 

197.37 

21 

22 

236.92 

227.73 

219.23 

211.37 

204.06 

197.26 

22 

23 

236.76 

227.58 

219.10 

211.24 

203.94 

197.15 

23 

24 

236.60 

227.43 

218.96 

211.11 

203.83 

197.04 

24 

25 

236.44 

227.29 

218.82 

210.99 

203.71 

196.93 

25 

26 

236.28 

227.14 

218.69 

210.86 

203.59 

196.82 

26 

27 

236.13 

226.99 

218.56 

210.74 

203.48 

196.71 

27 

28 

235.97 

226.85 

218.42 

210  61 

203.36 

196.60 

28 

29 

235.81 

226.70 

218.29 

210.49 

203.24 

196.49 

29 

30 

235.65 

226.55 

218.15 

210.36 

203.13 

196.39 

30 

31 

235.49  . 

226.50 

218.01 

210.25 

203.01 

196.28 

31 

32 

235.34 

226.26 

217.88 

210.11 

202.89 

196.17 

32 

33 

235.18 

226.12 

217.75 

209.99 

202.78 

196  06 

33 

34 

235.02 

225.97 

217.61 

209.86 

202.66 

195.95 

34 

35 

234.87 

225.83 

217.48 

209.74 

202.54 

195.84 

35 

36 

234.71 

225.68 

217.34 

209.61 

202.43 

195.74 

36 

37 

234.55 

225.54 

217.21 

209.49 

202.31 

195.63 

37 

38 

234.40 

225.40 

217.08 

209.37 

202.20 

195.52 

38 

°9 

234.24 

225.25 

216.94 

209.24 

202.08 

195.41 

39 

40 

234.08 

225.11 

216.81 

209.12 

201.97 

195.31 

40 

41 

233.93 

224.96 

216.68 

208.99 

201.85 

195.20 

41 

42 

233.77 

224.82 

216.54 

208.87 

201.74 

195.09 

42 

43 

233.62 

224.68 

216.41 

208.75 

201.62 

194.99 

43 

44 

233.46 

224.53 

216.28 

208.63 

201.51 

194.88 

44 

45 

233.31 

224.39 

216.15 

208.50 

201.40 

194  77 

45 

46 

233.15 

224.25 

216.02 

208.38 

201.28 

194.67 

46 

47 

233.00 

224.11 

215.88 

208.26 

201.17 

194.56 

47 

48 

232.84 

223.96 

215.75 

208.14 

201.05 

194.45 

48 

49 

232.69 

223.82 

215.62 

208.01 

200.94 

194.34 

49 

50 

232.54 

223.68 

215.49 

207.89 

200.83 

194.24 

50 

51 

232.38 

223.54 

215.36 

207.77 

200.71 

194.13 

51 

52 

232.23 

223.40 

215.23 

207.65 

200.60 

194.03 

52 

53 

962.08 

223.26 

215.10 

207.52 

200.49 

193.92 

53 

54 

231.92 

223.11 

214.96 

207.40 

200.37 

193.82 

54 

55 

231.77 

222.97 

214.83 

207.28 

200.  2  i 

193  71 

55 

56 

231.62 

222.83 

214.70 

207.16 

200.15 

193.61 

56 

57 

231.47 

222.69 

214.57 

207.04 

200.03 

193.50 

57 

58 

231.31 

222.55 

214.44 

206.92 

199.92 

193.40 

58 

59 

231.16 

222.41 

214.31 

206.80 

199.81 

193.29 

59 

60 

231.01 

222.27 

214.18 

206.68 

199.70 

193.19 

60 

65 


NATURAL   SINES    AND  COSINES. 


0° 

1° 

2° 

3° 

4° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

' 

0 

.00000 

One. 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

60 

1 

.00029  One. 

.01774 

.99984 

.03519 

.99938 

.05263 

.99861 

.07005 

.99754  59 

2 

.00058  One. 

.01803 

.99984 

.03548 

.99937 

.05292 

.99860 

.07034 

.99752  58 

3 

.00087  One. 

.01832 

.99983 

.03577 

.99936 

.05321 

.99858 

.07063 

.99750 

57 

4 

.00116!  One. 

.01862 

.99983 

.03606 

.99935 

.05350 

.99857 

.07092 

.99748 

56 

5 

.00145 

One. 

.01891 

.99982 

.03635 

.99934 

.05379 

.99855 

.07121 

.  99746  !  55 

6 

.00175 

One. 

.01920 

.99982 

.03664 

.99933 

.05408 

.99854 

.07150 

.99744!  54 

7 

.00204 

One. 

.01949 

.99981 

.03693 

.99932 

.05437 

.99852 

.07179 

.99742  53 

8 

.00233 

One. 

.01978 

.99980 

.03723 

.99931 

.05466 

.99851 

.07208 

.99740  52 

9 

.00262 

One. 

.02007 

.99980 

.03752 

.99930 

.05495 

.99849 

.07237 

.99738'  51 

10 

.00291 

One. 

.02036 

99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

.00320 

.99999 

.02065 

.99979 

.03810 

.99927 

.05553 

.99846 

.07295 

.99734 

49 

12 

.00349 

.99999 

.02094 

.99978 

.03839 

.99926 

.05582 

.99844 

.07324 

.99731 

48 

13 

.00378 

.99999 

.02123 

.99977 

.03868 

.99925 

.05611 

.  99842  ! 

.07353 

.99729 

47 

14 

.00407 

.99999 

.02152 

.99977 

.03897 

.99924 

.05640 

.99841  i 

.07382 

.99727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839! 

.07411 

.99725 

45 

16 

.00465 

.99999 

.02211 

.99976 

.03955 

.99922 

.05698 

.99838: 

.07440 

.99723 

44 

17 

.00495 

.99999 

.02240 

.99975 

.03984 

.99921 

.05727 

.99836 

.07469 

.99721 

43 

18 

.00524 

.99999 

.02269 

.99974 

.04013 

.99919 

.05756 

.99834 

.07498 

.99719 

42 

19 

.00553 

.99998 

.02298 

.99974 

.04042 

.99918 

.05785 

.998331 

.07527 

.99716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

.00611 

.99998 

.02356 

.99972 

.04100 

.99916 

.05844 

.99829 

.07585 

.99712 

39 

22 

.00640 

.99998 

.02385 

.99972 

.04129 

.99915 

.05873 

.99827 

.07614 

.99710 

38 

23 

.00669 

.99998 

.02414 

.99971 

.04159 

.99013 

.05902 

.99826 

.07643 

.99708 

37 

24 

.00698 

.99998 

.02443 

.99970 

.04188 

.99912 

.05931 

.99824; 

.07672 

.99705 

36 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822: 

.07701 

.99703 

35 

26 

.00756 

.99997 

.02501 

.99969 

.04246 

.99910 

.05989 

.99821 

.07730 

.99701 

34 

27 

.00785 

.99997 

.02530 

.99968 

.04275 

.99909 

.06018 

.99819! 

.07759 

.99699 

33 

28 

.00814 

.99997 

.02560 

.99967 

.04304 

.99907 

.06047 

.99817! 

.07788 

.99696 

32 

29 

.00844 

.99996 

.02589 

.99966 

.04333 

.99906 

.06076 

.99815 

.07817 

.99694 

31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

.99692 

30 

31 

.00902 

.99996 

.02647 

.99965 

.04391 

.99904 

.06134 

.99812  ! 

.07875 

.99689 

29 

32 

.00931 

.99996 

.02676 

.99964 

.04420 

.9D9C2 

.06103 

.99810 

.07904 

.99687 

28 

33 

.00960 

.99995 

.02705 

.99963 

.04449 

.99901 

.06192 

.99808 

.07933 

.99685 

27 

34 

.00989 

.99995 

.02734 

.99963 

.04478 

.99900 

.06221 

.99806 

.07962 

.99683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680 

25 

36 

.01047 

.99995 

.02792 

.99961 

.04536 

.99897 

.06279 

.99803 

.08020 

.99678 

24 

37 

.01076 

.99994 

.02821 

.99960 

.04565 

.99896 

.06308 

.99801! 

.08049 

.99676 

23 

38 

.01105 

.99994 

.02850 

.99959 

.04594 

.99894 

.06337 

.99799 

.08078 

.99673 

22 

39 

.01134 

.99994 

.02879 

.99959 

.04623 

.99893 

.06366 

.99797 

.08107 

.99671 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

.01193 

.99993 

.02938 

.99957 

.04682 

.99890 

.06424 

.99793 

.08165 

.99666 

19 

42 

.012221.99993 

.02967 

.99956 

.04711 

.99889 

.06453 

.99792 

.08194 

.99664 

18 

43 

.01251  .99992 

.02996 

.99955 

.04740 

.99888 

.06482 

.99790; 

.08223 

.99661 

17 

44 

.01280  .99992 

.03025 

.99954 

.04769 

.99886 

.06511 

.99788; 

.08252 

.99659 

16 

45 

.01309  .99991 

.03054 

.99953 

.04798 

.99885 

.06540 

.99786! 

.08281 

.99657 

15 

46 

.01338  .99991 

.03083 

.99952 

.04827 

.99883 

.06569 

.99784' 

.08310 

.99654 

14 

47 

.013671.99991 

.03112 

.99952 

.04856 

.99882 

.06598 

.99782 

.08339 

.99652 

13 

48 

.01396  .99990 

.03141 

.99951 

.04885 

.99881 

.06627 

.99780 

.08368 

.99649 

12 

49 

.01425 

.99990 

.03170 

.99950 

.04914 

.99879 

.06656 

.99778 

.08397 

.99647 

11 

50 

.01454 

.99989 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776 

.08426 

.99644 

10 

51 

.01483 

.99989 

.03228 

.99948 

.04972 

.99876 

.06714 

.99774 

.08455 

.99642 

9 

52 

.01513 

.99989 

.03257 

.99947 

.05001 

.99875 

.06743 

.99772 

.08484 

.99639 

8 

53 

.01542 

.99988 

.03286 

.99946 

.05030 

.99873 

.0677-3 

.99770, 

.08513 

.99637 

7 

54 

.01571 

.99988 

.03316 

.99945 

.05059 

.99872 

.06802 

.99768 

.08542 

.99635 

6 

55 

01600 

.99987 

.03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

.01629 

.99987 

.Oa374 

.99943 

.05117 

.99869 

.06860 

.99764 

.08600 

.99630 

4 

57 

.016581.99986 

.03403 

.99942 

.05146 

.99867 

.06889 

.99762 

.08629 

.99627 

3 

58 

.01  687  i.  99986 

.03432 

.99941 

.05175 

.99866 

.06918 

.99760; 

.08658 

.99625 

2 

59 

.01716 

.99985 

.03461 

.99940 

.05205 

.99864 

.06947 

.99758 

.08687 

99622 

1 

60 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

.  08716  j.  99619 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosm 

Sine 

Cosin 

Sine 

Cosin  i  Sine 

t 

89" 

88° 

*7° 

86° 

85' 

66 


NATURAL   SINES  AND   COSINES. 


/ 

5° 

6° 

7° 

8* 

9° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

~0 

.08716 

.99619 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

60 

1 

.08745 

.99617 

.10482 

.99449! 

.12216 

.99251 

.13946 

.99023 

.15672 

.98764!  59 

2 

.08774 

.99614 

'.10511 

.  99446  ! 

.12245 

.99248 

.13975 

.99019 

.15701 

.98^60  58 

3 

.08803 

.99612 

.10540 

.99443 

.12274 

.99244 

.14004 

.99015 

.15730 

.98755  5? 

4 

.08831 

.99609 

.10569 

.99440 

.12302 

.99240 

.14033 

.99011 

.15758 

.98751!  56 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

.157871.98746  55 

6 

.08889 

.99604 

.10626 

.99434 

.12360 

.99233! 

.14090 

.99002 

.15816  .98741  54 

7 

.08918 

.99602 

.10655 

.99431 

.12389 

.  99230  j 

.14119 

.98998 

.15845 

.98737  53 

8 

.08947 

.99599 

.10684 

.99428 

.12418 

.99226 

.14148 

.98994 

.15873 

.987321  52 

9 

.08976 

.99596 

.10713 

.99424 

.12447 

.99222 

.14177 

.98990 

.15902 

.98728 

51 

10 

.09005 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

.09034 

.99591 

.10771 

.99418 

.12504 

.99215 

.14234 

.98982 

.15959 

.98718 

4G 

12 

.09063 

.99588 

.10800 

.99415 

.12533 

.99211 

.14263 

.98978 

.15988 

.98714 

4t 

13 

.09092 

.99586 

.10829 

.99412 

.12562 

.99203] 

.14292 

.98973  .16017 

.98709 

4? 

14 

.09121 

.99583 

.10858 

.99409 

.12591 

.99204 

.14320 

.98969 

.16046 

.98704 

41 

15 

.09150 

.99580 

.10887 

.99406 

.12620 

.99200 

.14349 

.98965 

.16074 

.98700 

4.: 

16 

.09179 

.99578 

.10916 

.99402 

.12849 

.99197 

.14373 

.98961 

.161031.98695 

44 

17 

.09208 

.99575 

.10045 

.99399 

.12678 

.99193 

.14407 

.98957 

.161321.98690 

49 

18 

.09237 

.99572 

.10973 

.99393 

.12706 

.99189 

.14436 

.98953 

.16160L98686 

4i 

-19 

.09266 

.99570 

.11002 

.99393 

.127,35 

.991C6 

.14464 

.98948 

.16189  .98681 

41 

20 

.09295 

.99507 

.11031 

.99390 

.12764 

.99182 

.14493 

.98944 

.16218 

.98676 

41 

21 

.09324 

.99564 

.11060 

.99386 

.12793 

.99178 

.14522 

.98940 

.16246 

.98671 

3C 

22 

.09353 

.99562 

.11030 

.99333, 

.12822 

.90175 

.14551 

.98936 

.16275 

.98667 

3f- 

23 

.09382 

.99559 

.11118 

.99380 

.12351 

.99171 

.14580 

.98931 

.16304 

.98662 

37 

24 

.09411 

.99556 

.11147 

.99377 

.12330 

.99167 

.14608 

.98927 

.16333 

.  98657 

31 

25 

.09440 

.99553 

.11176 

.9D374 

.12908 

.99163 

.14637 

.98923 

.16361 

.98652  35 

26 

.09469 

.99551 

.11205 

.99370 

.12937 

.99160 

.14666 

.98919 

.16390 

.98648'  34 

27 

.09498 

.99548 

.11234 

.99367 

.12966 

.99156 

.14695 

.98914 

.16419 

.98643!  33 

28 

.09527 

.99545 

.11263 

.99364 

.12995 

.99152 

.14723 

.98910 

.16447 

.98638]  32 

29 

.09556 

.99542 

.11291 

.99360 

.13024 

.99148 

.14752 

.98006 

.16476 

.98633!  31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

.16505 

.98629 

3C 

31 

.09614 

.99537 

.11349 

.99354 

.13081 

.99141 

.14810 

.98897 

.16533 

.98624 

2C 

32 

.09642 

.99534 

1.11378 

.93351 

.13110 

.9oicr 

.14338 

.98893 

.16562 

.98619 

2J. 

33 

.09671 

.99531 

.11407 

.99347 

.13139 

.99133 

.14867 

.98889 

.16591 

.98614!  2? 

34 

.09700 

.99528 

.11436 

.93344 

.13168 

.99123 

.14896 

.98884 

.16620 

.98609  2C 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

.98604!  2* 

36 

.09758 

.99523 

1  .11494 

.99337 

.13226 

.99123 

.14954 

.98876 

1  .16677 

.98600!  24 

37 

.09787 

.99520 

.11523 

.99334 

.13254 

.99118 

.14982 

.98871 

.16706 

.98595  2? 

38 

.09816 

.99517 

.11552 

.99331 

.13283 

.99114 

.15011 

.98867 

.16734 

.98590  2S 

39 

.09845 

.09514 

.11580 

.93327 

.13312 

.99110 

.15040 

.98863 

.16763 

.98585  21 

40 

.09874 

.99511 

.11009 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

21 

41 

.09903 

.99508 

.11638 

.99320 

.13370 

.99102 

.15097 

.98854 

.16820 

.98575 

li 

42 

.09932 

.99500 

.110G7 

.99317 

.13399 

.99003 

.15126 

.98849  .16849 

.98570 

If 

43 

.09961 

.99503 

.11090 

.99314 

.13427 

.90094 

.15155 

.98845 

.168781.98565 

r 

44 

.09990 

.99500 

.11725 

.99310 

.13456 

.99091 

.15184 

.98841 

.16906  |.98561 

K 

45 

.10019 

.99497 

.11754 

•99307 

.13485 

.99087 

.15212 

.98836 

.169351.98556 

it 

46 

.10048 

.99494 

.11783 

.99303 

.13514 

.99083 

.152-41 

.98832 

.16964  L98551 

14 

47 

,10077 

.99491 

.11812 

.99300 

.13543 

.99079 

.15270 

.98827 

.16992 

.98546 

« 

48 

.10106 

.99488 

.11840 

.99297 

.13572 

.99075 

.15299 

.98823 

.17021 

.98541 

IS 

49 

.10135 

.99485 

.11869 

.99293 

.13600 

.99071 

.15327 

.98818 

.17050 

.98536 

11 

50 

.10164 

.99482 

.11898 

.99290 

.13629 

.99067 

.15356 

.98814 

.17078 

.98531 

K 

51 

.10192 

.99479 

.11927 

.99286 

.13658 

.99063 

.15385 

.98809 

.17107 

.98526 

1 

52 

.10221 

.99476 

.11956 

.99283 

.13687 

.99059 

.15414 

.98805 

.17136 

.98521 

| 

53 

.10250 

.99473 

.11985 

.99279 

.13716 

.99055 

.15442 

.98800 

.17164 

.98516 

54 

.10279 

.99470 

.12014 

.99276 

.13744 

.99051 

.15471 

.98796 

.17193 

.98511 

I 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

.15500 

.98791 

.17222|.98506 

56 

.10337 

.99464 

.12071 

.99269 

.13802 

.99043 

.15529 

.98787 

.17250  .98501 

4 

57 

10366 

.99461 

.12100 

.99265 

.13831 

.99039 

.15557 

.98782 

.  172791.  98496  i  i 

58 

.10395 

.99458 

.12129 

.99262 

.13860 

.99035 

.15586 

.98778 

.17308 

.  98491  i  S 

59 

.10424 

.99455 

.12158 

.99258 

.13889 

.99031 

.15615 

.98773  !  .17,336 

.98486'  ] 

60 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

.  1  7365  j.  984811  ( 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

Cosin  |  Sine 

/ 

84° 

83* 

829 

81° 

80° 

67 


NATURAL  SINES  AND  COSINES. 


10° 

11° 

12° 

13° 

14° 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

Sine 

Cosin 

' 

0 

.17365  .98481 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

60 

1 

.17393  .98476 

.19109 

.98157 

.20820 

.97809 

.22523 

.97430 

.24220 

.97023 

59 

2 

.17422  .98471 

.19138 

.98152 

.20848 

.97803 

.22552 

.97424 

.24249 

.97015 

58 

3 

.17451  .98466 

.19167 

.98146 

.20877 

.97797 

.22580 

.97417 

.24277 

.97008 

57 

4 

.17479  .98461 

.19195 

.98140 

.20905 

.97791 

.22608 

.97411 

.24305 

.97001 

56 

5 

.17508  .98455 

.19224 

.98135 

.20983 

.97784 

.22637 

.97404 

.24333 

.96994 

55 

6 

.17537  .98450 

.19252 

.98129 

.20962 

.97778 

.22665 

-07398 

.24362 

.96987 

54 

7 

.17565  .98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.  96980 

53 

8 

.17594  .98440 

.19309 

.98118 

.21019 

.97766 

.22722 

.97384 

.24418 

.96973 

52 

9 

.17623  .98435 

.19338 

.98112 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10 

.17651 

.98430 

.19366 

.98107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

.17680 

.98425 

.19395 

.98101 

.21104 

.97748 

.22807 

.97365 

.24503 

.96952 

49 

12 

.17708 

.98420 

.19423 

.98096 

.21132 

.97742 

.22835 

.97358 

.24531 

.96945 

48 

13 

.17737 

.98414 

.  19452 

.98C90 

.21101 

.97735 

.22863 

.97351 

.24559 

.  96937 

47 

14 

.17766 

.98409 

.19481 

.98084 

.21189 

.97729 

.22892 

.97345 

.24587 

.96930 

46 

15 

.17794 

.98404 

.19509 

.98079; 

.21218 

.97723 

.22920 

.97338 

.24615 

.96923 

45 

16 

.17823 

.98399 

.19538 

.98073 

.21246 

.97717 

.22948 

.97331 

.24644 

.96916 

44 

17 

.17852 

.98394 

.19566 

.98067 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

18 

.17880 

.98339 

.19595 

.980611 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19 

.17909 

.98383 

.19623 

.98056 

.21331 

.97698 

.23033 

.97311 

.24728 

.96894 

41 

20 

.17937 

.98378 

.19652 

98050 

.21360 

.97692 

.23062 

.97304 

.2-4756 

.96887 

40 

21 

.17966 

.98373 

.19680 

.98044 

.21388 

.97686 

.23090 

.97298 

.24784 

.96880 

39 

22 

.17995 

.98368 

.19709 

98039 

.21417 

.97680! 

.23118 

.97291 

.24813 

.96873 

38 

23 

.18023 

.98362 

.19737 

98033 

.21445 

.97673 

.23146 

.97284 

.24841 

.96866 

37 

24 

.18052 

.98357 

.19766 

98027 

.21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

25 

.18081 

.98352 

.19794 

98021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109 

.98347 

.19823 

98016 

.21530 

.97655 

.23231 

.97264 

.24925 

.96844 

34 

27 

.18138 

.98341 

.19851 

98010  1 

.21559 

.97648 

.23260 

.97257 

.24954 

.96837 

33 

28 

.18166 

.98a36 

.19880 

980041 

.21587 

.97642 

.23288 

.97251 

.24982 

.96829 

32 

29 

.18195 

.98331 

.19908 

97998 

.21016 

.97035 

.23316 

.97244 

.25010 

.96822 

31 

30 

.18224 

.98325 

.19937 

97992 

.21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

97987 

.21672 

.97623 

.23373 

.97230 

.25066 

.96807 

29 

32 

.18281 

.98315 

.19994 

97981 

.21701 

.97017; 

.23401 

.97223 

.25094  .96800 

28 

33 

.18309 

.98310 

.20022 

9T975| 

.21729 

.97611 

.23429 

.97217 

.25122 

.96793 

27 

34 

.18338 

.98304 

.20051 

979G9 

.21758 

.97604! 

.23458 

.97210 

.25151 

.96786 

26 

35 

.18367 

.98299 

.20079 

97983  i 

.21786 

.97593: 

.23486 

.97203 

.25179 

.96778 

25 

36 

.18395 

.98294 

.20108 

97958 

.21814 

.97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

.18424 

.98288 

.20136 

97952 

.21843 

.97535 

.23542 

.97189; 

.25235 

.96764 

23 

38 

.18452 

.98283! 

.20165 

97946 

.21871 

.97579 

.23571 

.97182 

.25263 

.96756 

22 

39 

.18481 

.98277 

.20193 

97940 

.21839 

.97573 

.23599 

.97176  .25291 

.96749 

21 

40 

.18509 

.98272 

.20222 

97934 

.21928 

.  97566  i 

.23627 

.97169  :  .25320  .96742 

20 

41 

.18538 

.98267 

.20250 

97928 

.21956 

.97560 

.23656 

.97162!  .253481.96734 

19 

42 

.18567 

.98261 

.20279 

97922 

.21985 

.97553 

.23684 

.97155  .25376 

.96727 

18 

43 

.18595 

.98256 

.20307 

97916 

.22013 

.97547 

.23712 

.97148  .25404 

.96719 

17 

44 

.18624 

.98250 

.20336 

97910 

.22041 

.97541 

.23740 

.97141  .25432 

.96712 

16 

45 

.18652 

.98245 

.20364 

97905 

.22070 

.97534 

.23769 

.97134-  .25460 

.96705 

15 

46 

.18681 

.98240 

.20393 

97899 

.22098 

.97528 

.23797 

.97127  .25488 

.96697 

14 

47 

.18710 

.98234 

.20421 

97893 

.22126 

.97521 

.23825 

.97120  !  .25516 

.96690 

13 

48 

.18738 

.98229 

.20450 

97887 

.22155 

.97515 

.23853 

.97113  i  .25545 

.96682 

12 

49 

.18767 

.98223) 

.20478 

97881 

.22183 

.97508 

.23882 

.97106  1  .25573 

.96675 

11 

50 

.18795 

.98218 

.20507 

97875 

.22212 

.97502 

.23910 

.97100 

.25601 

.96667 

10 

51 

.18824 

.98212 

.20535 

97869 

.22240 

.97496 

.23938 

.97093 

.25629 

.96660 

9 

52 

.18852 

.98207 

.20563 

97863 

.22268 

.97489 

.23966 

.97086 

.25657 

.96653 

8 

53 

.18881 

.98201  1  .20592 

.97857 

.22297 

.97483 

.23995 

.97079 

.25685 

.96645 

7 

54 

.18910 

.98196  1  .20620 

.97851 

.22325 

.97476 

.24023 

.97072 

.25713 

.96638 

6 

55 

.18838 

.98190 

.20649 

.97845 

.22353 

.97470 

.24051 

.970651 

.257411.96630 

5 

56 

.189671.98185 

.20677 

.97839 

.22382 

.97463 

.24079 

.97058| 

.257691.96623 

4 

57 

.189951.98179 

.20706 

.97833 

.22410 

.97457 

.24108 

.97051 

.25798.96615  3 

58 

.19024 

.98174 

.20734 

.97827 

.22438 

.97450 

.24136 

.97044 

.25826  .96608  2 

59 

.19052 

.98168 

.20763 

.97821 

.22467 

.97444 

.24164 

.97037 

.25854  .96600j  1 

60 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.  25882  ,.  96593|  0 

f 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

f 

79° 

78° 

77°    i    "76°    !    75° 

-NATURAL   SINES   AND 


15° 

16° 

17° 

18° 

19° 

/ 

Sine  1  Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

~0 

.25882  .96593 

.27564  .96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

60 

1 

.25910 

.96585 

.27592 

.96118 

.29265 

.956221  .30929 

.95097 

.32584 

.94542 

59 

2 

.25938 

.96578 

.27620 

.96110 

.29293 

.95613 

,  .30957 

.95088 

.32612 

.94533 

58 

3 

.25966 

.96570 

.27648 

.96102 

.29321 

.95605 

1.30985 

.95079 

.32639 

.94523 

57 

4 

.259941.96562 

.27676 

.96094 

.29348 

.95596 

1  .34012 

.95070 

.32667 

.94514 

56 

5 

.26022  .96555 

.27704 

.96086 

.29376 

.95588  !  .31040 

.95061 

.32694 

.94504 

55 

6 

.26050  .96547 

.27731 

.96078 

.29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7 

.260791.96540 

.27759 

•96070 

.29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.26107J.  96532 

.27787 

.96062 

.29460 

.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135  .96524 

.27815 

.96054 

.29487 

.95554 

.31151 

.95024 

.32804 

.94466 

51 

10 

.26163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191 

.96509 

.27871 

.96037 

.29543 

.95536 

.31206 

.95000 

.32859 

.94447 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.95528 

.31233 

.94997 

.32887 

.94438 

48 

13 

.26247 

.96494 

.27927 

.96021 

.29599 

.95519 

.31201 

.94988 

1  .32914 

.94428 

47 

14 

.26275 

.96485 

.27955 

.96013 

.29026 

.95511 

.31289 

.94979 

.32942 

.94418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31310 

.94970 

.32969 

.94409 

45 

1G 

.26331 

.96471! 

.28011 

.95997 

.29682 

.95493 

.31344 

.94961 

.32997 

.94399 

44 

17 

.26359 

.96463 

.28039 

.95989 

.29710 

.95485 

.31372 

.94952 

.33024 

.94390 

43 

18 

.2638? 

.96456 

.28067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.26415 

.96448 

.28095 

.95972 

.29765 

.95467 

.31427 

.94933 

.33079 

.94370 

41 

20 

.26443 

.  96440  ( 

.28123 

.95964 

.29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40 

21 

.26471 

.96433 

.28150 

.95956 

.29821 

.95450 

.31482 

.94915 

.33134 

.94351 

39 

22 

.26500 

.96425 

.28178 

.95948 

.29849 

.95441 

.31510 

.94900 

.33161 

.94342 

38 

23 

.26528 

.96417 

.28206 

.95940 

.20876 

.95433 

.31537 

.94897 

.33189 

.94332 

37 

24 

.26556 

.96410 

.28234 

.95931 

.29904 

.95424 

.31505 

.94888 

.a3216 

.94322 

36 

25 

.28584 

.96402 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313 

35 

26 

.26612 

.96394 

.28290 

.95915 

.29900 

.95407 

.31620 

.94809 

.33271 

.94303 

34 

27 

.26640 

.96386 

.28318 

.95907 

.29987 

.95398 

.31048 

.94800 

.33298 

.94293 

33 

28 

.26668 

.96379 

.28346 

.95898 

.30015 

.95389 

.31075 

.94851 

.33326 

.94284 

32 

29 

.28696 

.96371 

.28374 

.95890 

.30043 

.95380 

.31703 

.94842 

.33353 

.94274 

31 

30 

.20724 

.96.363 

.28402 

.95882 

.30071 

.95372 

.31730 

.94832 

.33381 

.94264 

30 

31 

.20752 

.96355 

.28429 

.95874 

.30098 

.95363 

.31750 

.94823 

.33408 

.94254 

29 

82 

.26780 

.9034? 

.28457 

.95805 

.30120 

.95354 

.31780 

.94814 

.33436 

.94245 

28 

S3 

.26808 

.90340 

.28485 

.95857 

.30154 

.95345 

.31813 

.94805 

.33463 

.94235 

27 

34 

.26836 

.96332 

.28513 

.95849 

.30182 

.95337 

.31841 

.94795 

.33490 

.94225 

26 

£5 

.26864 

.96324 

.28541 

.95841 

.30209 

.95328 

.31808 

.94786 

.33518 

.94215 

25 

36 

.26892 

.96316: 

.28569 

.95832 

.30237 

.95319 

.31890 

.94777 

.33545 

.94206 

24 

37 

.26920 

.98303 

.28597 

.95824 

.30205 

.95310 

.31923 

.94768 

.33573 

.94196 

23 

38 

.26948 

.90301 

.28625 

.95816 

.30292 

.95301 

.31951 

.94758 

.33000 

.94186 

22 

39 

.26976 

.98293 

.28652 

.95807 

.30320 

.95293 

.31979 

.94749 

.33627 

.94176 

21 

40 

.27004 

.96285 

.28680 

.95799 

.30348 

.95284 

.3200C 

.94740 

.33655 

.94167 

20 

41 

.27032 

.96277 

.28708 

.95791 

.30376 

.95275 

.32034 

.94730 

.33682 

.94157 

19 

42 

.27060 

.96269 

.28736 

.95782 

.30403 

.95203 

.320G1 

.94721 

.33710 

.94147 

18 

43 

.27088 

.96261 

.28764 

.95774 

.30431 

.95257 

.32080 

.94712 

.33737 

.94137 

17 

44 

.27116 

.96253 

.28792 

.95760 

.30459 

.95248 

.32110 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.95757 

.30486 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

*  28847 

.95749 

.30514 

.95231 

.32171 

.94684 

.33819 

.94108 

14 

47 

.27200 

.96230 

.'28875 

.95740 

.30542 

.95222 

.32199 

.94674 

.33846 

.94098 

13 

48 

.27228 

.96202 

.28903 

.95732 

.30570 

.95213 

.32227 

.94665 

.33874 

.94088 

12 

49 

.27256 

.96214' 

.28931 

.95724 

.30597 

.95204 

.32254 

.94656 

33901 

.94078 

11 

50 

.27284 

.96206 

.28959  .95715 

.30625 

.95195 

.32282 

.94640' 

.33929 

.94068 

10 

51 

.27312 

.96198 

.28987  .95707 

.30653 

.95186 

.32309 

.94637 

.33956 

.94058 

9 

52 

.27340  .96190, 

.29015  .95698 

.30680 

.9517? 

.32337 

.94627 

.33983 

.94049 

8 

53 

.27368  .96182 

.29042  .95690 

.30708 

.  95108  i 

.32364 

.94618 

.34011 

.94039 

7 

54 

.27396  .96174 

.29070 

.95681 

.3073(3 

.95159 

.32392 

.94609 

.34038 

.94029 

6 

55 

.27424  .96166 

.29098 

.95673 

.30763 

.95150 

.32419 

.94599 

.34065 

.94019 

5 

56 

.27452  .96158 

.29126  .95664 

.30791 

.95142 

.32447 

.94590 

.34093 

.94009 

4 

57 

.27480  .96150 

.29154  .95656 

.30819 

.95133 

.32474 

.93580 

.34120 

.93999 

3 

58 

.27508;.  96142 

.29182  .95647 

.30846 

.95124 

.32502 

.94571 

.34147 

.93989 

2 

59 

.27536  .96134 

.29209 

.95639 

.30874 

.95115 

.32529 

.94561 

.34175 

.93979 

1 

60 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

.34202 

.93969 

0 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

i 

74° 

73° 

72° 

71° 

70° 

69 


NATURAL  SINES  AND  COSINES. 


20° 

21°   1 

22° 

23° 

24° 

Sine 

Cosin 

Sine  !  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

0 

.34202 

.93969 

.358371.93358] 

.37461 

.92718 

.39073 

.92050 

.40674  .913551  60 

1 

.34229 

.93959 

.35864 

.93348 

.37488 

.92707 

.39100 

.92039 

.40700  .91343  59 

2 

.34257 

.93949 

.35891 

.93337 

.37515 

.92697 

.39127 

.92028 

.40727 

.91331  58 

3 

.34284 

.93939 

.35918 

.93327 

.37542 

.92686 

.39158 

.92016 

.40753 

.91319  57 

4 

.34311 

.93929 

.35945 

.93316 

.37569 

.92675 

.39180 

.92005 

.40780 

.  91307  i  56 

5 

.34339 

.93919 

.35973 

.93306 

.37595 

.92664 

.39207 

.91994 

.40806 

.91295  55 

6 

.34366 

.93909 

.36000 

.93295 

.37622 

.92653 

.39234 

.91982 

.40833 

.91283  54 

7 

.34393 

.93899 

.36027 

.93285 

.37649 

.92642 

.39260 

.91971 

.40860 

.91272;  53 

8 

.34421 

.93889 

.36054 

.93274 

.37676 

.92631 

.39287 

.91959 

.40886 

.91260  52 

9 

.34448 

.93879 

.36081 

.93264 

.37703 

.92620 

.39314 

.91948 

.40913  .91248  51 

10 

.34475 

.93869 

.36108 

.93253 

.37730 

.92609 

.39341 

.91936 

.40939  .91236 

50 

11 

.34503 

.93859 

.36135 

.93243 

.37757 

.92598 

.39367 

.91925 

.40966 

.91224 

49 

12 

.34530 

.93849 

.36162 

.93232 

.37784 

.92587 

.39394 

.91914 

.40992 

.91212 

48 

13 

.34557 

.93839 

.36190 

.93222 

.37811 

.92576 

.39421 

.91902 

.41019 

.91200 

47 

14 

.34584 

.93829 

.36217  .93211 

.37838 

.92565 

.39448 

.91891 

.41045 

.91188 

46 

15 

.34612 

.93819 

.36244  '.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

45 

16 

.34639 

.93809 

.362711.93190 

.37892 

.92543 

.39501 

.91868 

.41098 

.91164 

44 

17 

.34666 

.93799 

.36298:.  93180 

.37919 

.92532 

.39528 

.91856 

.41125 

.91152 

43 

18 

.34694 

.93789 

.363251.93169 

.37946 

.92521 

.39555 

.91845 

.41151 

.911401  42 

19 

.34721 

.93779 

.36352  .93159 

.37973 

.92510 

.39581 

.91833 

.41178 

.91128  41 

20 

.34748 

.93769 

.363791.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116  40 

21 

.34775 

.93759 

.36406  .93137 

.38026 

.92488 

.39635 

.91810 

.41231 

.91104 

39 

22 

.34803 

.93748 

.36434 

.93127 

.38053 

.92477 

.39661 

.91799 

.41257 

.91092!  38 

23 

.34830 

.93738 

.36461 

.93116 

.38080 

.92466 

.39688 

.91787 

.41284 

91080  37 

24 

.34857 

.93728 

.36488 

.93106 

.38107 

.92455 

.39715 

.91775 

.41310 

.910681  36 

25 

.34884 

.93718 

.36515 

.93095 

.38134 

.92444 

.39741 

.91764 

.41337 

.91056  35 

26 

.34912 

.93708 

.36542 

.93084 

.38161 

.92432 

.39768 

.91752 

.41363 

.91044  34 

27 

.34939 

.93698 

.36569 

.93074 

.38188 

.92421 

.39795 

.91741 

.41390 

.91032!  33 

28 

.34966 

.93688 

.36590 

.93063 

.38215 

.92410 

.39822 

.91729 

.41416 

.91020;  32 

29 

.34993 

.93677 

.36623 

.93052 

..C8241 

.92399 

.39848 

.91718 

.41443 

.91008  31 

30 

.35021 

.93667 

.36650 

.93042 

.38268 

.92388 

.39875 

.91706 

.41469 

:  90996 

30 

81 

.35048 

.9365? 

.36677 

.93031 

.38295 

.92377 

.39902 

.91694 

.41496 

.90984 

29 

32 

.35075 

.93647 

.36704 

.93020 

.38322 

.92360 

.39928 

.91683 

.41522 

.90972;  28 

33 

.35102 

.93637 

.36731 

.93010 

.38349 

.92355 

.39955 

.91671 

.41549 

.90960  27 

34 

.35130 

.93626 

.36758 

.92999 

.38376 

.92343 

.39982 

.91660 

.41575 

.90948  26 

35 

.35157 

.93616 

.36785 

.92988 

.38403 

.92332 

.40008 

.91648 

.41602 

.90936|  25 

36 

.35184 

.93606 

.36812 

.92978 

.38430 

.92321 

.40035 

.91636 

.41628 

.90924  24 

37 

.35211 

.93596 

.36839 

.92967 

.38456 

.92310 

.40062 

.91625 

.41655 

.90911 

23 

38 

.35239 

.93585 

.36867 

.92956 

.38483 

.92299 

.40088 

.91613 

.41681 

.90899  22 

39 

.35266 

.93575 

.36894 

.92945 

.38510 

.92287 

.40115 

.91601 

.41707 

.90887  21 

40 

.35293 

.93565 

.36921 

.92935 

.38537 

.92276 

.40141 

.91590 

.41734!  90875  go 

41 

.35320 

.93555 

.36948 

.92924 

.38564 

.92265 

.40168 

.91578 

.41760  .90863!  19 

42 

.35347 

.93544 

.36975 

.92913 

.38591 

.92254 

.40195 

.91566 

.41787  .90851  18 

43 

.35375 

.93534 

.37002 

.92902 

.38617 

.92243 

.40221 

.91555 

.41813  .90839  17 

44 

.35402 

.93524 

.37029 

.92892 

.38644 

.92231 

.40248 

.91543 

.41840 

.90826'  16 

45 

.35429 

.93514 

.37056 

.92881 

.38671 

.92220 

.4027'5 

.91531 

.41866 

.90814 

15 

46 

.35456 

.93503, 

.37083 

.92870 

.38698 

.92209 

.40301 

.91519 

.41892 

.90802 

14 

47 

.35484 

.93493: 

.37110 

.92859 

.38725 

.92198 

.40328 

.91508 

.41919 

.90790 

13 

48 

.35511 

.93483; 

.37137 

.92849 

.38752 

.92186 

.40355 

.91496 

.41945 

.90778 

12 

49 

.35538 

.93472 

.37164 

.92838 

.38778 

.92175 

.40381 

.91484 

.41972 

.90766 

11 

50 

.35565 

.93462 

.37191 

.92827 

.38805 

.92164 

.40408 

.91472 

.41998 

.90753 

10 

51 

.35592 

.93452 

.37218 

.92816 

.38832 

.92152 

.40434 

.91461 

.42024 

.90741 

9 

52 

.35619 

.93441 

.37245 

.92805 

.38859 

.92141 

.40461  .91449 

.42051 

.90729 

8 

53 

.35647 

.93431 

.37272 

.92794 

.38886 

.92130 

.40488  .91437 

.42077 

.90717 

7 

54 

.35674 

.93420 

.37299 

.92784 

.38912 

.92119 

.40514!.  91425 

.42104 

.90704 

6 

55 

.35701 

.93410! 

.37326 

.92773 

.38939 

.92107 

.40541  j.  91414 

.42130 

.90692 

5 

56 

.35728  .93400 

.37353 

.92762 

.38966  .92096 

.405671.91402 

.42156 

.90680 

4 

57 

.35755  .93389; 

.37380 

.92751 

.389931.92085 

.40594  .91390 

.42183 

.90668 

3 

58 

.35782 

.93379; 

.37407 

.92740 

.39020 

.92073 

.40621 

.91378 

.42209 

.90655 

2 

59 

.35810 

.93368 

.37434 

.92729 

.39046 

.92062 

.40647 

.91366 

.42235 

.90643 

1 

60 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674  .91355 

.42262 

.90631 

0 

f 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

"Sine 

Cosin  !  Sine 

Cosin 

Sine 

/ 

69° 

68° 

67° 

66°. 

65° 

70 


NATURAL  SINES  AND  COSINES. 


25° 

26° 

27° 

28°   : 

29° 

/ 

Sine  iCosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  ,  Cosin 

~o 

.422621.90631 

.43837 

".  89879 

.45399 

.89101 

.46947i.88295 

.48481 

.87462 

60 

1 

.42288  '.9061  8 

.43863 

.89867 

.45425 

.89087 

.46973 

.882811 

.48506 

.87448 

59 

2 

.42315  .90606 

.43889 

.89854 

.45451 

.89074 

.46999 

.882671 

.48532 

.87434 

58 

3 

.42341  .90594 

.43916 

.89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367  .90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56 

5 

.42394  .90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420  .90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.424461.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473  .90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.424991.90520 

.44072 

.89764 

.45632 

.88981 

.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.44124 

.89739 

.45684 

.88955 

.47229 

.88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90408 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

.44385 

.89610 

.45942 

.88822 

.47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45963 

.83803 

.47511 

.87993 

.49040 

.87150 

38 

23 

.42867 

.90346 

.44437 

.89581 

.45994 

.83795 

.47537 

.87979 

.49065 

.87136 

37 

24 

'.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.46097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

28 

.42999 

.90284 

.44503 

.89519 

.46123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.93271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87896 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

.44646 

.89480 

.46201 

.88688 

.47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.46223 

.83074 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.44698 

.89454 

.46252 

.88661 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90208' 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196- 

.44750 

.89428 

.46304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

.43209 

.90183 

.44776 

.89415 

.46330 

.88620 

.47869 

.87798 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88607 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.46381 

.88593 

.47920 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.46407 

.88580 

.47946 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90108 

.44932 

.89337 

.46484 

.88539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.46510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44984 

.89311 

.46536 

.88512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46561 

.88499 

.48099 

.87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.48124 

.87G59 

.49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.86791 

13 

48 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

.48175 

.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46664 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981 

.45192 

.89206 

.46742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86690 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

0 

/ 

Cosin 

Sine 

Cosin 

"Sine" 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

64° 

63° 

1   62° 

61° 

60° 

71 


NATURAL  SINES  ANB  COSINES. 


30° 

31° 

32° 

33° 

34° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

0 

.50000 

.86603 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

60 

1 

.50025 

.86588 

.51529 

.85702 

.53017 

.84789; 

.54488 

.83851 

.55943 

.82887 

59 

2 

.50050 

.86573 

.51554 

.85687 

.53041 

.84774 

.54513 

.83835 

.55968 

.82871 

58 

3 

.50076 

.86559 

.51579 

.85672 

.53066 

.84759 

.54537 

.83819 

.55992 

.82855;  57 

4 

.50101 

.86544 

.51604 

.85657 

.53091 

.84743 

.54561 

.83804 

.56016 

.82839  56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

.56040 

.82822  55 

6 

.50151 

.86515 

.51653 

.85627 

.53140 

.84712 

.54610 

.83772 

.56064 

.82806  54 

7 

.50176 

.86501 

.51678 

.85612 

.53164 

.84697! 

.54635 

.83756 

.56088 

.82790  53 

8 

.50201 

.86486 

.51703 

.85597 

.53189 

.84681 

.54659 

.83740 

.56112 

.82773  52 

9 

.50227 

.86471 

.51728 

.85582 

.53214 

.846661 

.54683 

.83724 

.56136 

.827571  51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.  84650  \ 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635' 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53288 

.84619 

.54756 

.83676 

.56208 

.82708  48 

13 

.50327 

.86413 

.51828 

.85521 

.53312 

.84604 

.54781 

.83660 

.56232 

.82692'  47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588 

.54805 

.83645 

.56256 

.82675  46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83629 

.56280 

.82659  |  45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854 

.83613 

.56305 

.82643  44 

17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626  43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610  42 

19 

.50478 

.86325 

.51977 

.85431 

.53460 

.84511 

.54927 

.83565 

.56377 

.82593  41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.56401 

.82577  40 

21 

.50528 

.86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

.82561  39 

22 

.50553 

.86281 

.52051 

.85385 

.53534 

.84464 

.54999 

.83517 

.5G449 

.82544  38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84448 

.55021 

.83501 

.56473 

.82528!  37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55040 

.83485 

.56497 

.82511136 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.  82495  i  35 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83453 

.56545 

.82478  34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.56569 

.82462  33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.  82446  i  32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.551G9 

.83405 

.5GG17 

.82429  31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413  30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.56665 

.82396!  29 

32 

.50804 

.86133 

.52299 

.85234 

.53779 

.84308 

.55242 

.83356 

.56689 

.82380!  28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55206 

.83340 

.56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277 

.55201 

.83324 

.56730 

.82347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.83308 

.66760 

.82330  25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245 

.55339 

.83292 

.56784 

.82314 

24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.55363 

.83276 

.56808 

.82297 

23 

38 

.50954 

.86045 

.52448 

.85142 

.53926 

.84214 

.55383 

.83260 

.56832 

.82281!  22 

39 

.50979 

.86030 

.52473 

.85127 

.53951 

.84198 

.55412 

.83244 

.56856 

.82264  21 

40 

.51004 

.86015 

.52498 

.85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

.52522 

.85096 

.54000 

.84167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.54024 

.84151 

.55484 

.83195 

.56928 

.82214 

18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509 

.83179 

.56952 

.82198 

17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533 

.83163 

.56976 

.82181 

16 

45 

.51129 

;  85941 

.52621 

.85035 

.54097 

.84104 

.55557 

.83147 

.57000 

.82165 

15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.83131 

.57024 

.82148 

14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.57047 

.82132 

13 

48 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098 

.57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082 

.57095 

.82098 

11 

50 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.55678 

.83066 

.57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050 

.57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54269 

.83994 

.55726 

.83034 

.57167 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017 

.57191 

.82032 

7 

54 

.51354 

.85806 

.52844 

.84897 

.54317 

.83962 

.55775 

.83001 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.54366 

.83930 

.55823 

.82969 

.57262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.55847 

.82953 

.57286 

.81965 

3 

58 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.57310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820 

.54440 

.83883 

.55895 

.82920 

.57334 

.81932 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

.57358 

.81915 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosiu 

~SimT 

/ 

59° 

58° 

57° 

56° 

55° 

NATURAL  SINES  AND  COSINES. 


35° 

36° 

37° 

38° 

39° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

o 

.57358 

7819151 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

60 

1 

.57381 

.  81899  i 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696 

59 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829! 

.61612 

.78765 

.62977 

.77678 

58 

3 

.57429 

.81865 

.58849 

.808501 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660 

57 

4 

.57453 

.81848: 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.63022 

.77641 

56 

5 

.57477 

.81832, 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711! 

.63045 

,77623 

55 

6 

.57501 

.81815 

.58920 

.80799 

.60321 

.79758 

.61704 

.78694) 

.63068 

.77605 

54 

.57524 

.81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548 

.81782; 

.58967 

.80765 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.77550 

51 

10  .57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531 

50 

11  .57619 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77513 

40 

12  1  .57643 

.81714 

.59061 

.  80696  i 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13  .57667 

.81698 

.59084 

.80679^ 

.60483 

.79635 

.61864 

.78568 

.632251.77476 

47 

14 

.57691 

.81681 

.59108 

.80662 

.60506 

.79618| 

.61887 

.78550 

.63248J.77458 

46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532 

.63271 

.77439 

45 

16 

.57738 

.81647 

.59154 

.80627 

.60553 

.79583 

.61932 

.78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

.59178 

.80610 

.60576 

.79565 

.61955  .78496 

.63316  .77402 

43 

18 

.57786 

.81614 

.59201 

.  80593  i 

.60599 

.79547 

.61978  .78478 

.63338  .77384 

42 

19 

.57810 

.81597 

.59225 

.80576 

.60622 

.79530! 

.620011.78460 

.633611.77366 

41 

20 

.57833 

.81580 

.59248 

.80358 

.60645 

.79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

.57857 

.81563 

.59272 

.80541 

.60668 

.79494 

.62046 

.78424 

.63406 

.77329 

39 

22 

.57881 

.81546 

.59295 

.805241 

.60691 

.79477 

.62069 

.78405 

.63428 

.77310 

38 

23 
24 

.57904 
.57928 

.81530 
.81513 

.59318 
.59342 

.80507 
.80489 

.60714 
.60738 

.79459 
.79441 

.62092 
.62115 

.78387 
.78369 

.63451  .77292 
.63473  .77273 

37 
36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255 

35 

26 

.57976 

.81479 

.59389 

.80455 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

27 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388 

.62183 

.78315 

.63540 

.77218 

33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.62206 

.78297 

.63563 

.77199 

32 

29 

.58047 

.81428 

.59459 

.80403 

60853 

.79353 

.62229 

.78279 

.63585 

.77181 

31 

30 

.58070 

.81413 

.59482 

.80386 

.60876 

.79335  .62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368 

.60899 

.79318  .62274 

.78243 

.63630 

.77144 

29 

32 

.58118 

.81378 

.59529 

.80351 

.60922 

.79300 

.62297 

.78225 

.63653 

.77125 

28 

33 

.58141 

.81361 

.59552 

.80334 

.60945 

.79282 

.62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81344 

.59576 

.80316 

.60968 

.79264 

.623421.78188 

.63698 

.77088 

26 

35 

.58189 

.81327 

.59599 

.80299 

.60991 

.79247 

.62365 

.78170 

.637201.77070 

25 

36 

.58212 

.81310 

.59622 

.80282 

.61015 

.79229 

.62388 

.78152 

.63742 

.77051 

24 

37 

.58236 

.81293 

.59646 

.80264 

.61038 

.79211 

.62411 

.78134 

.63765 

.77033 

23 

38 

.58260 

.81276 

.59669 

.802471 

.61061 

.79193 

.62433 

.78116 

.63787 

.77014 

22 

39 

.58283 

.81259 

.59693 

.80230 

.61084 

.79176 

.62456 

.78098 

.63810 

.76996 

21 

40 

.58307 

.81242 

.59716 

.80212 

.61107 

.79158 

.62479 

.78079 

.63832 

.76977 

20 

41 

.58330 

.81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061 

.63854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122 

.62524 

.78043 

.63877 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921 

17 

44 

.58401 

.81174 

.59809 

.80143 

.61199 

.79087 

.62570 

.78007 

.63922 

.76903 

16 

45 

.58425 

.81157 

.59832 

.80125 

.61222 

.79069 

.62592 

.77988 

.63944 

.76884 

15 

46 

.58449 

.81140 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966 

.76866 

14 

47 

.58472 

.81123 

.59879 

.80091 

.61268 

.79033 

.62638 

.77952 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.80073 

.61291 

.79016 

.62660 

.77934 

.64011 

.76828 

12 

49 

.58519 

.81089 

.59926 

.80056 

.61314 

.78998 

.62683 

.77916 

.64033 

.76810 

11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791 

10 

51 

.58567 

.81055 

.59972 

.80021 

.61360 

.78962 

.62728 

.77879 

.64078 

.76772 

9 

52 

.58590 

.81038 

.59995 

.80003 

.61383 

.78944 

.62751 

.77861 

.64100 

.76754 

8 

53 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843 

.64123 

.76735 

7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824, 

.64145 

.76717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

.77806' 

.64167J.  76698 

5 

56 

.58084 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788; 

.64190  .76679  4 

57 

.58708 

.80953 

.60112 

.79916 

.61497 

.78855 

.62864 

.777691 

.64212  .766611  3 

58 

.58731 

.80936 

.60135 

.79899 

.61520 

.78837 

.62887 

.  77751  1 

.64234  .76642 

2 

59 

.58755 

.80919 

.60158 

.79881 

.61543 

.78819 

.62909 

.777331 

.64256  .76623 

1 

60 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

.64279  .76604 

o 

i 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  |  Sine 

7 

54° 

53° 

52° 

51* 

50° 

73 


NATURAL  SINES  AND  COSINES. 


40° 

41° 

42° 

43' 

44° 

Sine  Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine 

Cosin 

' 

0 

.64279 

.76604 

.6500b 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934  60 

1 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914  59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894  58 

3 

.64346 

.76548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873  57 

4 

.64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853  56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833  55 

6 

.64412 

.76492 

.65738 

.75356 

.67043 

.74198 

.68327 

.73016 

.69591 

.71813  54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792  53 

8 

.64457 

.76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69633 

.71772  52 

9 

.64479 

.76436 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752  51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732  50 

11 

.64524 

.76398 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711  49 

12 

.64546 

.76380 

.65869 

.75241 

.67172 

.74080 

.68455 

.72897 

.69717 

.716911  48 

13 

.64568 

.7G361 

.65891 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

.70342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

45 

16 

.64635 

.76304 

.65956 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64657 

.76286 

.65978 

.75146 

.67280 

.73983 

.68561 

.72797 

.69821 

.71590 

43 

18 

.64679 

.76267 

.66000 

.75126 

.67301 

.739G3 

.68582 

.72777 

.69842 

.71569 

42 

19 

.64701 

.76248 

.66022 

.75107 

.67323 

.73944 

.68G03 

.72757 

.69862 

.71549 

41 

20 

.64723 

.76229, 

.66044 

75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

75089 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192 

.66088 

75050 

.67387 

.73885 

.68GGO 

.72697 

.69925 

.71488 

38 

23 

.64790 

.76173 

.66109 

75030 

.67409 

.738G5 

.68683 

.72677 

.69946 

.71468 

37 

24 

.64812 

.76154 

.66131 

75011 

.67430 

.7'3846 

.68709 

.72657 

.69966 

.71447 

36 

25 

.64834 

.76135 

.66153 

74992 

.67452 

.73820 

.68730 

.7'2637 

.699871.71427 

35 

26 

.64856 

.76116 

.66175 

74973 

.67473 

.73800 

.68751 

.72617 

.70008  .71407 

34 

27 

.64878 

.76097 

.66197 

74953 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386 

33 

28 

.64901 

76078 

.68218 

74934 

.G7516 

.73767 

.68793 

.72577 

.70049 

.71366 

32 

29 

.64923 

76059 

.66240 

74915 

.G7538 

.73747 

.68814 

.72557 

.70070 

.71345 

31 

30 

.64945 

76041 

.66262 

74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325 

30 

31 

.64967 

76022 

.66284 

74876 

.67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989 

76003 

.66306 

74857 

.67602 

.73688 

.68878 

.72497 

70132 

.71284 

28 

33 

.65011 

759841 

.66327 

74838 

.67623 

.73669 

.68899 

.72477 

.70153 

.71264 

27 

34 

.65033 

75965 

.66349 

74818 

.67645 

.73649 

.68920 

.72457 

.70174 

.71243 

26 

35 

.65055 

75946 

.66371 

74799 

.67666 

.73629 

.68941 

.72437 

.70195 

.71223 

35 

36 

.65077 

75927 

.66393 

74780 

.67688 

.73610 

.68962 

.72417 

.70215 

.71203 

24 

37 

.65100 

75908 

.66414 

74760 

.67709 

.73590 

.68983 

.72397 

.70236 

.71182 

23 

38 

.65122 

75889 

.66436 

74741 

.67730 

73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.65144 

75870 

.66458 

74722 

.67752 

73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

75851 

.66480 

74703 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

75832 

.66501 

74683 

.67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

75813 

.66523 

74664 

.67816 

.73491 

.69088 

.72297 

.70339 

.71080 

18 

43 

.65232 

75794 

.66545 

74644 

.67837 

.73472 

.69109 

.72277 

.70360 

.71059 

17 

44 

.65254 

75775 

.66566 

74625 

.67859 

.73452 

.69130 

.72257 

.70381 

.71039 

16 

45 

.65276 

75756 

.66588 

74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

.65298 

75738 

.66610 

74586 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998 

14 

47 

.65320 

75719 

.66632 

74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978 

13 

48 

.65342 

75700 

.66653 

74548 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65364 

75680 

.66675 

74528 

.67965 

.73353 

.69235 

.72156 

.70484 

.70937 

11 

50 

.65386 

75661 

.66697 

74509  | 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

.65408 

75642 

.66718 

74489! 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896 

9 

52 

.65430 

75623 

.66740 

74470  i 

.68029 

.73294 

.69298 

.72095 

.70546 

.70875 

8 

53 

.65452 

75604 

.66762 

74451 

.68051 

.73274 

.69319 

72075 

.70567 

.70855 

7 

54 

.65474 

75585 

.66783 

74431 

.68072 

.73254 

.69340 

72055 

.70587 

.70834 

6 

55 

.65496 

.75566 

.66805 

74412 

.68093 

.73234 

.69361 

72035 

.70608  .70813 

5 

56 

.65518 

.75547! 

.66827 

.743921 

.68115 

.73215 

.69382 

.72015 

.706281.70793 

4 

57 

.65540 

.75528 

.66848 

.74373! 

.68136 

.73195 

.69403 

.71995 

.70649 

.70772 

3 

58 

.65562 

.75509 

.66870 

74353 

.68157 

.73175 

.69424 

.71974 

.70670 

.70752 

2 

59 

.65584 

.75490 

.66891 

.  74334  l 

.68179 

.73155 

.69445 

.71954 

.70690 

.70731 

1 

60 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

.70711 

.70711 

0 

' 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

f 

49° 

48° 

47° 

46° 

45° 

NATURAL  TANGENTS  AND  COTANGENTS. 


0° 

1°             I!            2°            |l            3° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang  I  Cotang 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811    (-0 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755    59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1064 

.05299 

18.8711 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645 

55 

0 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.6367 

.05503 

18.1708 

51 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

40 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

10 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

190.984 

.02269 

44.0661 

.04016 

24.8978 

.05706 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

22 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1C57 

.05883 

16.9990 

38 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

24 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

2(; 

.00756 

132.219 

,02502 

39.9G55 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

28 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.00058 

16.5075 

32 

29 

.00844 

118.540 

.02589 

38.G177 

.04337 

23.0577 

.00087 

16.4283 

31 

30 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7686 

.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

34 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

86 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

37 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

88 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

39 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02968 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

10 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

40 

.01338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

47 

.01367 

73.1390 

.03114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.00817 

14.6685 

0 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

50 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

f 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  |    Tang 

/ 

89° 

88°           \\           87° 

86° 

75 


NATURAL  TANGENTS  AND  COTANGENTS. 


4° 

5° 

6° 

7° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

00 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

6 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.12426 

8.04756 

55 

6 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

8 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07'285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9882 

.108G3 

9.20516 

.12033 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10803 

9.18C28 

.12602 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12092 

7.87895 

40 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86064 

45 

16 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

10 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

30 

22 

.07636 

13.0958 

.09394 

10.6450 

.11153 

8.90227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.C9453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

30 

25 

.07724 

12.9469 

.C9482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

35 

20 

.07753 

12.8981 

.C9511 

10.5136 

.11270 

8.86862 

.13047 

7.66466 

:  ;  1 

27 

.07783 

12.8496 

.C9541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

88 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.C96CO 

10.4172 

.11864 

8.79964 

.13136 

7.61287 

31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

31 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

20 

32 

.07929 

12.6124 

.C9GC8 

10.3224 

.11452 

8.73172 

.12224 

7.50176 

33 

.07958 

12.5660 

.C9717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.13284 

7.52806 

20 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806 

23 

38 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.C9893 

10.1080 

.11659 

8.57718 

.13432 

7.44509 

21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

T.  53402 

.13491 

7.41240 

10 

42 

.08221 

12.1632 

.C0981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.C8251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11800 

8.47007 

.13580 

7.36389 

10 

45 

.08309 

12.0346 

.10069 

9.93101 

.11830 

8.44896 

.13609 

7.34786 

15 

40 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13609 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

5 

56 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.121CO 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11.5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

85° 

84° 

83° 

82° 

NATURAL  TANGENTS  AND  COTANGENTS. 


8° 

9° 

10° 

11° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

60 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

3 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

56 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

I   .17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96823 

1   .16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.04267 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10G64 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

J2 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

38 

\  r'3 

.14737 

6.78564 

.16525 

6.05143 

.18323 

5.45751 

I   .20133 

4.96690 

37 

* 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945 

36 

» 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

.26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

;28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18504 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.878S2 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.83151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17004 

5.83114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.  86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77286 

10 

01 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

3 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

2 

59 

.15809 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137 

1 

C>0 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

81° 

80° 

79° 

78° 

77 


NATURAL  TANGENTS  AND  COTANGENTS. 


12° 

13° 

14° 

1           15° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24964 

4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.215£9 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

12 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

40 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

41 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.01839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

2:3 

.21925 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

88 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27513 

3.63461 

37 

24 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

36 

25 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

20 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

33 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

20 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

3  60588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

2<i 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

34 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

20 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

30 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

•3.58160 

21 

37 

.22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

2:} 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.2617'2 

3^82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

10 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

10 

43 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

1  .28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

10 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22843 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

3.48741 

0 

f 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

77° 

76° 

75° 

74° 

78 


-NATURAL  TANGENTS  AND  COTANGENTS. 


16° 

17° 

18° 

19° 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang   j  Cotang 

Tang 

Cotang 

0 

.28675 

3.48741 

.30573 

3.27085 

.82492     3.07768 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

G 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

53 

8 

.28927 

3.45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9 

.28958 

3.45327 

.30860 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2.87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

13 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.03260 

.34922 

2.86356 

45 

10 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02667 

.34987 

2.85822 

43 

18 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.85020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35052 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.3313G 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.331C9 

3.01489 

.35118 

2.84758 

39 

22 

.29368 

3.40502 

.31274 

3.19752 

.S3201 

3.01196 

.35150 

2.84494 

38 

23 

.29400 

3.40136 

.31306 

3.19426 

.332C3 

3.00903 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.19100 

.332C6 

3.COG11 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

26 

.29495 

3.39042 

.31402 

3.18451 

.sasso 

3.00028 

.35281 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.18127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.314G6 

3.17804 

.33395 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

3.17481 

.33427 

2.C9158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

32 

.29685 

3.36875 

.31594 

3.16517 

.33524 

2.88292 

.35477 

2.81870 

28 

33 

.29716 

3.36516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.36158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

35 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

3G 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.356C8 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.33686 

2.96858 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.06573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31818 

3.14288 

.33751 

2.962C8 

.85707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31011 

3.13341 

.33848 

2.95437 

.358C5 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.358G8 

2.79033 

17 

44 

.300G5 

3.32614 

.31978 

3.12713 

.33913 

2.9487'2 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

46 

.30123 

3.31914 

.32042 

3.12087 

.33978 

2.94309 

.35937 

2.78269 

14 

47 

.301  GO 

3.315G5 

.32074 

3.11775 

.34010 

2.94028 

.35909 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.114G4 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3.308G8 

.32f39 

3.11153 

.34075 

2.934G8 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10843 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34308 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32460 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

66 

.30573 

3.27085 

.32492 

3.077G8 

.34433 

2.90421 

.36397 

2.74748 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotanj 

Tang 

Cotang 

Tang 

/ 

73° 

72° 

71°           II           70° 

79 


NATURAL  TANGENTS  AND  COTANGENTS. 


20° 

21° 

22° 

23° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

; 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

i 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

] 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

i 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

6 

.36595 

2.732G3 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

\ 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

.36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

.40809 

2.45043 

.42860 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

18 

.36991 

2.70335 

.38988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.3902.2 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55G08 

1   .41149 

2.43019 

.43205 

2.31456 

38  i 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42618 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

35 

26 

.37256 

2.68414 

.39257 

2.54734 

.412G5 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2.67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66939 

.39458 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.39492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

36 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710 

23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528 

22 

39 

.37687 

2.65342 

.39694 

2.51929 

.41728 

2.39645 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806 

18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41805 

2.38863 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39862 

2.50864 

41899 

2.38668 

.43966 

2.27447 

16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.38084 

.44071 

2.26909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

2.63021 

.40031 

2.49807 

.42070 

2.37697 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10, 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2  48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2.24780 

1 

60 

38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

f 

69° 

68° 

67° 

66°           | 

80 


NATURAL  TANGENTS  AND  COTANGENTS. 


24° 

25°           |!           26° 

27° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.44523 

2.24604 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

60 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

1.96120 

59 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44662 

2.23902 

.4677'2 

2.13801 

.48917 

2.04426 

.51099 

1.95698 

56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

6 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

1.95277 

53 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12832 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49 

S3 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

1C 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

1.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746 

42 

19 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

1.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

1.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

1.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

2(5 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946  • 

1.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

1.92371 

32 

29 

.45538 

2.19599 

.47662 

2.C9811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

1.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

:>>•• 

.45643 

2.19092 

.47769 

2.09341 

.43931 

2.00277 

.52131 

1.91826 

28 

33 

,45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

1.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

1.99695 

.52279 

1.91282 

24 

37    .45819 

2.18251 

.47948 

2.08560 

.50113 

1.99550 

.52316 

1.91147 

23 

38    .45854 

2.18084 

.47984 

2.03405 

.50149 

1.99406 

.52353 

1.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

1.99261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

1.99116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

1.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

1.98828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07'630 

.50331 

1.98684 

.52538 

1.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

.50368 

1.98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

1.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

1.98253 

.52650 

1.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

1.98110 

.52687 

1.8C801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

1.97966 

.52724 

1.89667 

12 

49 

.46242 

2.16255 

.48378 

2.06706 

.50550 

1.97823 

.52761 

1.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

1.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

1.97538 

.52836 

1.89266 

9 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

1.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

1.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

1.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

1.96969 

.52985 

1.88734 

5 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

1.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

|   .50843 

1.96685 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

1   .50879 

1.96544 

.53096 

1.88337 

2 

59 

.46595 

2.14614 

.48737 

2.05182 

.50916 

1.96402 

.53134 

1.88205 

1 

GO 

.46631 

2.14451 

.48773 

2.05030 

1    .50953 

1.96261 

.53171 

1.88073 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

i  Cotang 

Tang 

Cotang 

Tang 

f 

65° 

64° 

63° 

62° 

81 


NATURAL  TANGENTS  AND  COTANGENTS. 


2 

go 

2 

9° 

3 

0° 

3 

i° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.53171 

.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

.87941 

.55469 

.80281 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

.87'809 

.55507 

.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

.87677 

.55545 

.80034 

.57851 

1.72857 

.60205 

1.66099 

57 

4 

.53320 

.87546 

.55583 

.79911 

.57890 

1.72741 

.60245 

1.65990 

56 

5 

.53358 

.87415 

.55621 

.79788 

.57929 

1.72625 

.60284 

1.65881 

55 

6 

.53395 

.87283 

.55659 

.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

.87152 

.5569? 

.79542 

.58007 

1.72393 

.60364 

1.65663 

53 

8 

.53470 

.87021 

.55736 

.79419 

.58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

1.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

1.86760 

.55812 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55964 

1.78685 

.58279 

1.71588 

.60642 

1.64903 

46 

15 

.53732 

1.86109 

.5G003 

1.78563 

I   .58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.00721 

1.64687 

44 

17 

.53807 

1.85850 

.50079 

1.78319 

.58396 

1.71244 

.60761 

1.64579 

43 

18 

.53844 

1.85720 

.50117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

19 

.53882 

1.85591 

.50156 

1.78077 

.50474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

1.85462 

.56194 

1.77955 

!   .58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

.53995 

1.85204 

.50270 

1.77713 

.58591 

1.70673 

.60960 

1.64041 

38 

23 

.54032 

1.85075 

.50309 

1.77592 

.58631 

1.70500 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

.50347 

1.77471 

.58670 

1.70446 

.61040 

1.63826 

36 

25 

.54107 

1.84818 

.50385 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

20 

.54145 

1.84689 

.56424 

1.77230 

.58743 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

1.84561 

.50462 

1.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

1.84433 

.50501 

1.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

29 

.54258 

1.84305 

.5C5S9 

1.70869 

.58865 

1.69879 

.61240 

1.63292 

31 

30 

.54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.69653 

.61320 

1.63079 

29 

32 

.54371 

1.83922 

.50654 

1.70510 

.58983 

1.69541 

.61360 

1.62972 

28 

33 

.54409 

1.83794 

.50693 

1.7G390 

.59022 

1.69428 

.61400 

1.62866 

27 

34 

.54446 

1.83667 

.56731 

1.76271 

.59061 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

1.83540 

.50769 

1.76151 

.5910JL 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62548 

24 

37 

.54560 

1.83286 

.56846 

1.75913 

.59179 

.68979 

.61561 

1.62442 

23 

38 

.54597 

1.83159 

.56885 

1.75794 

.59218 

.68806 

.61601 

1.62336 

22 

39 

.54635 

1.83033 

.50923 

1.75675 

.59258 

.68754 

.61641 

1.62230 

21 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

.68643 

.61681 

1.62125 

20 

41 

.54711 

1.82780 

.57000 

1.75437 

.59336 

.68531 

.61721 

1.62019 

19 

42 

.54748 

1.82654 

.57039 

1.75319 

.59376 

.68419 

.61761 

1.61914 

18 

43 

.54786 

1.82528 

.57078 

1.75200 

.59415 

.68308 

.61801 

1.61808 

17 

44 

.54824 

1.82402 

.57116 

1.75082 

.59454 

.68196 

.61842 

1.61703 

16 

45 

.54862 

1.82276 

.57155 

1.74964 

.59494 

.68085 

.61882 

1.61598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

.59533 

.67974 

.61922 

1.61493 

14 

47 

.54938 

1.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.67752 

.62003 

1.61283 

12 

49 

.55013 

1.81774 

.57309 

1.74492 

.59651 

1.67641 

.62043 

1.61179 

11 

50 

.55051 

1.81649 

.57348 

1.74375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

.55089 

1.81524 

.57386 

1.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

1.81399 

.57425 

1.74140 

.59770 

1.67309 

.62164 

1.60865 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

.59809 

1.67198 

.62204 

1.60761 

7 

54 

.55203 

1.81150 

.57503 

1.73905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59888 

1.66978 

.62285 

1.60553 

5 

56 

.55279 

1.80901 

.57580 

1.73671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.73555 

.59967 

1.66757 

.62366 

1.60345 

3 

58 

.55355 

1.80653 

.57657 

1.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

1.80529 

.57696 

1.73321 

.60046 

1.66538 

.62446 

1.60137 

1 

60 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

.62487 

1.60033 

0 

/ 

Cotang 

Tang    j 

Cotang 

Tang 

Cotang  | 

Tang 

Cotang 

Tang 

/ 

6 

_          ,  —  j 

6 

0°            1 

5 

3° 

5 

3° 

NATURAL  TANGENTS  AND  COTANGENTS. 


32° 

33° 

34° 

35° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.62487 

1.60033 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

00 

1 

.62527 

1.59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

59 

2 

.62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

.62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

50 

5 

.62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55 

6 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

52 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

0 

.62973 

1.58797 

.65438 

1.52816  ! 

.67900 

1.47146 

.70542 

1.41759 

48 

3 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

.63055 

1.58593 

.65521 

1.52622 

.68045 

1.40902 

.70629 

1.41584 

40 

15 

.63095 

1.58490 

.65563 

1.52525  1 

.68088 

1.46870 

.70673 

1.41497 

45 

16 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17 

.63177 

1.58286 

.65646 

1.52332 

.68173 

1.46686 

.70760 

1.41322 

48 

18 

.63217 

1.58184 

.65688 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.52139 

.68258 

1.46503 

.70848 

1.41148 

41 

SO 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1  .  57778 

.65854 

1.51850 

.08386 

1.46229 

.70979 

1.40887 

:;S 

23 

.63421 

1.57676 

.65896 

1.51754 

.68429 

1.40137 

.71023 

1.40800 

37 

24 

.63462 

1.57575 

.65938 

1.51058 

.68471 

1.4G046 

.71066 

1.40714 

3i  J 

25 

.63503 

1.57474 

.65980 

1.51502 

.68514 

1.45955 

.71110 

1.40627 

35 

26 

.63544 

1.57372 

.66021 

1.51,466  I 

.68557 

1.45864 

.71154 

1.40540 

•;4 

27 

.63584 

1.57271 

.06063 

1.51370 

.68600 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.06105 

1.51375 

.68642 

1.45682 

.71242 

1.40367 

:5-3 

29 

.63666 

1.57069 

.00147 

1.51179 

.68685 

1.45592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.00189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1  .45410 

.71373 

1.40109 

29 

32 

.63789 

1.50707 

.GG27'2 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

:JS 

33 

.63830 

1.50667 

.06314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27 

34 

.63871 

1.56566 

.00356 

1.50702 

.68900 

1.45139 

.71505 

1.39850 

20 

35 

.63912 

1.56466 

.66398 

1.50007 

.68942 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.56366 

.66440 

1.50512 

.08985 

1.44958 

.71593 

1.39679 

21 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.G4035 

1.56165 

.66524 

1.50322 

.09071 

1.44778 

.71681 

1.39507  122 

39 

.64076 

1.56065 

.66506 

1.50228 

.69114 

1.44688 

.71725 

1.39421  i21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.GGG92 

1.49944 

.69243 

1.44418 

.71857 

1.39165  il8 

43 

.64240 

1.55666 

.06734 

1.49849 

.69286 

1.44329 

.71901 

1.39079    17 

44 

.64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994  116 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

.71990 

1.38909  115 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824  '14 

47 

.64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738    13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653    12 

49 

.64487 

1.55071 

.06986 

1.49284 

.69545 

1.43792 

.72167 

1.38568  ill 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

51 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

52 

.64610 

1.54774 

i    .07113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53 

.64652 

1.54675 

.07155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64693 

1.54576 

!   .07197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

55 

.64734 

1.54478 

i   .67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

5 

56 

.64775 

1.54379 

.07282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

57 

.64817 

1.54281 

.07324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

5! 

.64858 

1.54183 

!   .67366 

1.48442 

.69934 

1.42992 

.  72565 

1.37807 

2 

59 

.64899 

1.54085 

i   .67409 

1.48349 

.69977 

1.42903 

.72610 

1.37723 

1 

60 

.64941 

1.53986 

i   .67451 

1.48256 

.70021 

1.42815 

.72054 

1.37638 

0 

Cotang  j    Tang 

Cotang 

Tang 

Cotang 

Tang 

i  Cotang 

^fang 

9 

57°                       56° 

55°            i           54° 

88 


NATURAL  TANGENTS  AND  COTANGENTS. 


36° 

37° 

38° 

39° 

Tang 

Cotang 

Tang  |  Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.72654 

1.37638 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

60 

1 

.72699 

1.37554 

.75401 

1.32624 

.78175 

1.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134  ! 

.75629 

1.32224 

.78410 

1.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

1.22904 

52 

9 

.73055 

1.30)883  i 

.75767 

1.31984 

.78551 

1.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

1.22685 

49 

12 

.73183 

1.8G633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

1.22321 

44 

17 

.73413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

1.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

1.22176 

42 

19 

.73503 

1.30051 

.76226 

1.31190 

.79022 

1.26546 

.81808 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

1.21959 

39 

22 

.73637 

1.35802 

.7'6364 

1.30952 

.79104 

1.26319 

.82044 

1.21886 

38 

23 

.73681 

1.35719 

.76410 

1.S0873 

.79212 

1.20244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

.76686 

1.30401 

.79496 

1.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

1.21238 

29 

32 

.74083 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

1.21166 

28 

33 

.74131 

1.34896 

.76871 

1.30087 

.70686 

1.25492 

.82580 

1.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

1.24375 

.83317 

1.20024  112 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

1  .  19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

1.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28379 

.80738 

1.23858 

.83662 

1.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

1.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

1.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

1.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

1.23563 

.83860 

1  .  19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

1.19175 

0 

/ 

Cotang 

Tang 

Cotang 

Tang      Cotang 

Tang 

;  Cotang     Tang 

t 

53° 

52°                       51° 

50° 

84 


NATURAL  TANGENTS  AND  COTANGENTS. 


4 

D° 

4 

L° 

4 

2° 

4 

3° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.83910 

1.19175 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237" 

60 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

.10996 

.93306 

1.07174 

59 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

.10931 

.93360 

1.07112 

58 

3 

.84059 

1.18964 

.87082 

1.14834 

.90199 

.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

.10802 

.93469 

1.06987 

56 

5 

.84158 

1.18824 

.87184 

1.14699 

.90304 

.10737 

.93524 

1.06925 

55 

6 

.84208 

1.18754 

.87236 

1.14632 

.90357 

.10672 

.93578 

1.06862 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

.10607 

.93633 

1.06800 

53 

8 

.84307 

1  .  18614 

.87338 

1.14498 

.90463 

.10543 

.93688 

1.06738 

52 

9 

.84357 

1.18544 

.87389 

1.14430 

.90516 

.10478 

.93742 

1.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

.10414 

.93797 

1.06613 

50 

11 

.84457 

1.18404 

.87492 

.14296 

.90621 

.10349 

.93852 

1.06551 

49 

12 

.84507 

1.18334 

.87543 

.14229 

.90674 

.10285 

.93906 

1.06489 

4H 

13 

.84556 

1  .  18264 

.87595 

.14162 

.90727 

.10220 

.93961 

1.06427 

47 

14 

.84606 

1.18194 

.87646 

.14095 

.90781 

.10156 

.94016 

1.06365 

40 

15 

.84656 

1.18125 

.87698 

.14028 

.90834 

.10091 

.94071 

1.06303 

45 

16 

.84706 

1.18055 

.87749 

.13961 

.90887 

.10027 

.94125 

1.06241 

41 

17 

.84756 

1  .  17986 

.87801 

1.13894 

.90940 

.09963 

.94180 

1.06179 

43 

18 

.84806 

1.17916 

.87852 

1.13828 

.90993 

.09899 

.94235 

1.06117 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046 

.09834 

.94290 

1.06056 

41 

20 

.84906 

1.17777 

.87955 

1.13694 

.91099 

.09770 

.94345 

1.05994 

40 

2i 

.84956 

1.17708 

.88007 

1.13627 

.91153 

1.09706 

.94400 

1.05932 

39 

22 

.85006 

1.17638 

.88059 

1.13561 

.91206 

1.09642 

.94455 

1.05870 

88 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

1.09578 

.94510 

1.05809 

87 

24 

.85107 

1  .  17500 

.88162 

1.13428 

.91313 

1.09514 

.94565 

1.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91366 

1.09450 

.94620 

1.05685 

36 

26 

.85207 

1.17361 

.88265 

1.13295 

.91419 

1.09386 

.94676 

1.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13228 

.91473 

1.09322 

.94731 

1.05562 

33 

28 

.85308 

1.17223 

.88369 

1.13162 

.91526 

1.09258 

.94786 

1.05501 

:52 

29 

.85358 

1.17154 

.88421 

1.13096 

.91580 

1.09195 

.94841 

1.05439 

31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

1.09131 

.94896 

1.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

1.09067 

.94952 

1.C5317 

29 

32 

.85509 

1.16947 

.88576 

1.12897 

.91740 

1.09003 

.95007 

1.05255 

28 

33 

.85559 

1.16878 

.88628 

1.12831 

.91794 

1.08940 

.95062 

.  .05194 

27 

34 

.85609 

1.16809 

.88680 

1.12765 

.91847 

1.08876 

.95118 

.05133 

20 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

.08813 

.95173 

.05072 

•35 

36 

.85710 

1.16672 

.88784 

1.12633 

.91955 

.08749 

.95229 

.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

.08686 

.95284 

.04949 

23 

38 

.85811 

1.16535 

.88888 

1.12501 

.92062 

1.08622 

.95340 

.04888 

22 

39 

.85862 

1.16466 

.88940 

1.12435 

.92116 

1.08559 

.95395 

.04827 

Ul 

40 

.85912 

1.16398 

.88992 

1.12369 

.92170 

1.08496 

.95451 

.04766 

•20 

41 

.85963 

1.16329 

|  .89045 

1.12303 

.02224 

1.08432 

.95506 

.04705 

10 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

1.08369 

.95562 

.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

1.08306 

.95618 

.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

1.08243 

.95673 

.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439 

1.08179 

.95729 

.04461 

15 

46 

.86216 

1.15987 

.89306 

1.11975 

.92493 

1.08116 

.95785 

.04401 

It 

47 

.86267 

1.15919 

.89358 

1.11909 

.92547 

1.08053 

.95841 

.04340 

18 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

1  -07990 

.95897 

.04279 

12 

49 

.86368 

1.15783 

.89463 

1  11778 

.92655 

1.07927 

.95952 

.04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

1.07864 

.96008 

.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11648 

.92763 

1.07801 

.96064 

.04097 

9 

52 

.86521 

1.15579 

.89620 

1.11582 

.92817 

1.07738 

.96120 

.04036 

8 

53 

.86572 

1.15511 

.89672 

1.11517 

.92872 

1.07676 

.96176 

.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

1.07613 

.96232 

.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

1.07550 

.96288 

.03855 

5 

56 

.86725 

1.15308 

.89830 

1.11321 

.93034 

1.07487 

.96344 

.03794 

4 

57 

.86776 

1.15240 

.89883 

1.11256 

.93088 

1.07425 

.96400 

.03734 

3 

58 

.86827 

1.15172 

.89935 

1.11191 

.93143 

1.07362 

.96457 

.03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

1.07299 

.96513 

1.03613 

1 

60 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

.96569 

1.03553 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

• 

4 

t9° 

4 

8° 

4 

7° 

4 

6° 

85 


NATURAL  TANGENTS  AND  COTANGENTS. 


t 

4 

4° 

^ 

14° 

1 

4 

4° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03553 

60 

20 

.97700 

1.02355 

40 

(40 

.98843 

1.01170 

30 

1 

.96625 

1.03493 

59 

21 

.97756 

1.02295 

39 

41 

.98901 

:  .01112 

u 

2 

.96681 

1.03433 

58 

22 

.97813 

1.02236 

88 

i42 

.98958 

.01053 

18 

3 

.96738 

1.03372 

57 

23 

.97870 

1.02176 

37 

43 

.99016 

.00994 

17 

4 

.96794 

1.03312 

56 

24 

.97927 

1.02117 

88 

44 

.99073 

.00935 

16 

5 

.96850 

1.G3252 

55 

25 

.97984 

1.02057 

,35 

45 

.99131 

.00876 

15 

6 

.96907 

1.03192 

54 

26 

.98041 

1.01998 

84 

I  46 

.99189 

.00818 

14 

7 

.96963 

1.03132 

53 

27 

.98098 

1.01939 

33 

47 

.99247 

:  .00759 

13 

8 

.97020 

1.03072 

52 

28 

.98155 

1.01879 

32 

48 

.99304 

.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

1.01820 

81 

i4D 

.99362 

:  .00642 

11 

10 

.97133 

1.02952 

50 

30 

.98270 

1.01761 

30 

50 

.99420 

.00583 

10 

11 

.97189 

1.02892 

49 

31 

.98327 

1.01702 

29 

51 

.99478 

.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

1.01642 

28 

52 

.99536 

.00467 

8 

13 

.97302 

1.02772 

47 

33 

.98441 

1.01583 

27 

53 

.99594 

.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

1.01524 

20 

54 

.99652 

:  .00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

1.01465 

25 

55 

.99710 

.00291 

5 

16 

.97472 

1.02593 

44 

86 

.98613 

1.01406 

24 

56 

.99768 

.00233 

4 

1? 

.97529 

1.02533 

43 

37 

.98671 

1.01347 

3s 

57 

.99826 

.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

1.01288 

22 

58 

.99884 

.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

1.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

60 

1.00000 

1.00000 

0 

Cotang 

Tang 

Cotang 

Tang 

/ 

Cotang 

Tang 

/ 

4 

5° 

4 

5° 

4 

5° 

NATURAL  VERSED  SINES  AND   EXTERNAL  SECANTS. 


0° 

1° 

2° 

3° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.00000 

.00000 

.00015 

.00015 

.00061 

.00061 

.00137 

.00137 

0 

1 

.00000 

.00000 

.00016 

.00016 

.00062 

.00062 

.00139 

00139 

1 

2 

.00000 

.00000 

.00016 

.00016 

.00063 

.00063 

.00140 

.00140 

2 

3 

.00000 

.00000 

.00017 

.00017 

.00064 

.00064 

.00142 

.00142 

3 

4 

.00000 

.00000 

.00017 

.00017 

.00065 

.00065 

.00143 

.00143 

4 

5 

.00000 

.00000 

.00018 

.00018 

.00066 

.00066 

.00145 

.00145 

5 

6 

.00000 

.00000 

.00018 

.00018 

.00067 

.00067 

.00146 

.00147 

6 

7 

.00000 

.00000 

.00019 

.00019 

.00068 

.00068 

.00148 

.00148 

7 

8 

.00000 

.00000 

.00020 

.00020 

.00069 

.00069 

.00150 

.00150 

8 

9 

.00000 

.00000 

.00020 

.00020 

.00070 

.00070 

.00151 

.00151 

9 

10 

.00000 

.00000 

.00021 

.00021 

.00071 

.00072 

.00153 

.00153 

10 

11 

.00001 

.00001 

.00021 

.00021 

.00073 

.00073 

.00154 

.00155 

11 

12 

.00001 

.00001 

.00022 

.00022 

.00074 

.00074 

.00156 

.00156 

12 

13 

.00001 

.00001 

.00023 

.00023 

.00075 

.00075 

.00158 

.00158 

13 

14 

.00001 

.00001 

.00023 

.00023 

.00076 

.00076 

.00159 

.00159 

14 

15 

.00001 

.00001 

.00024 

.00024 

.00077 

.00077 

.00161 

.00161 

15 

16 

.00001 

.00001 

.00024 

.00024 

.00078 

.00078 

.00162 

.00163 

16 

17 

.00001 

.00001 

.00025 

.00025 

.00079 

.00079 

.00164 

.00164 

17 

18 

.00001 

.00001 

.00026 

.00026 

.00081 

.00081 

.00166 

.00166 

18 

19 

.00002 

.00002 

.00026 

.00026 

.00082 

.00082 

.00168 

.00168 

19 

20 

.00002 

.00002 

.00027 

.00027 

.00083 

.00083 

.00169 

.00169 

20 

21 

.00002 

.00003 

.00028 

.00028 

.00084 

.00084 

.00171 

.00171 

21 

22 

.00002 

.00002 

.00028 

.00028 

.00085 

.00085 

.00173 

.00173 

22 

23 

.00002 

.00002 

.00029 

.00029 

.00087 

.00087 

.00174 

.00175 

23 

24 

.00002 

.00002 

.00030 

.00030 

.00088 

.00088 

.00176 

.00176 

24 

25 

.00003 

.00003 

.00031 

.00031 

.00089 

.00089 

.00178 

.00178 

25 

23 

.00003 

.00003 

.00031 

.00031 

.00090 

.00090 

.00179 

.00180 

26 

27 

.00003 

.00003 

.00032 

.00032 

.00091 

.00091 

.00181 

.00182 

27 

28 

.00003 

.00003 

.00033 

.00033 

.00093 

.00093 

.00183 

.00183 

28 

23 

.00004 

.00004 

.00034 

.00034 

.00094 

.00094 

.00185 

.00185 

29 

30 

.00004 

.00004 

.00034 

.00034 

.00095 

.00095 

.00187 

.00187 

30 

31 

.00004 

.00004 

.00035 

.00035 

.00096 

.00097 

.00188 

.00189 

31 

32 

.00004 

.00004 

.00036 

.00036 

.00098 

.00098 

.00190 

.00190 

32 

S3 

.00005 

.00005 

.00037 

.00037 

.00099 

.00099 

.00192 

.00192 

33 

34 

.00005 

.00005 

.00037 

.00037 

.00100 

.00100 

.00194 

.00194 

34 

35 

.00005 

.00005 

.00038 

.00038 

.00102 

.00102 

.00196 

.00196 

35 

3G 

.00005 

.00005 

.00039 

.00039 

.00103 

.00103 

.00197 

.00198 

36 

37 

.00006 

.00006 

.00040 

.00040 

.00104 

.00104 

.00199 

.00200 

37 

38 

.00006 

.00006 

.00041 

.00041 

.00106 

.00106 

.00201 

.00201 

38 

39 

.00006 

.00006 

.00041 

.00041 

.00107 

.00107 

.00203 

.00203 

39 

40 

.00007 

.00007 

.00042 

.00042 

.00108 

.00108 

.00205 

.00205 

40 

41 

.00007 

.00007 

.00043 

.00043 

.00110 

.00110 

.00207 

.00207 

41 

42 

.00007 

.00007 

.00044 

.00044 

.00111 

.00111 

.00208 

.00209 

42 

43 

.00008 

.00008 

.00045 

.00045 

.00112 

.00113 

.00210 

.00211 

43 

44 

.00008 

.00008 

.00046 

.00046 

.00114 

.00114 

.00212 

.00213 

44 

45 

.00009 

.00009 

.00047 

.00047 

.00115 

.00115 

.00214 

.00215 

45 

46 

.00009 

.00009 

.00048 

.00048 

.00117 

.00117 

.00216 

.00216 

46 

47 

.00009 

.00009 

.00048 

.00048 

.00113 

.00118 

.00218 

.00218 

47 

48 

.00010 

.00010 

.00049 

.00049 

.00119 

.00120 

.00220 

.00220 

48 

49 

.00010 

.00010 

.00050 

.00050 

.00121 

.00121 

.00222 

.00222 

49 

50 

.00011 

.00011 

.00051 

.00051 

.00122 

.00122 

.00224 

.00224 

50 

51 

.00011 

.00011 

.00052 

.00052 

.00124 

.00124 

.00226 

.00226 

51 

52 

.00011 

.00011 

.00053 

.00053 

.00125 

.00125 

.00228 

.00228 

52 

53 

.00012 

.00012 

.00054 

.00054 

.00127 

.00127 

.00230 

.00230 

53 

54 

.00012 

.00012 

.00055 

.00055 

.00128 

.00128 

.00232 

.00232 

54 

55 

.00013 

.00013 

.00056 

.00056 

.00130 

.00130 

.00234 

.00234 

55 

56 

.00013 

.00013 

.00057 

.00057 

.00131 

.00131 

.00236 

.00236 

56 

57 

.00014 

.00014 

.00058 

.00058 

.00133 

.00133 

.00238 

.00238 

57 

58 

.00014 

.00014 

.00059 

.00059 

.00134 

.00134 

.00240 

.00240 

58 

59 

.00015 

.00015 

.00060 

.00060 

.00136 

.00136 

.00242 

.00242 

69 

60 

.00015 

.00015 

.00061 

.00061 

.00137 

.00137 

.00244 

.00244 

60 

NATURAL  VERSED  SINES   AND  EXTERNAL  SECANTS 


/ 

4° 

5° 

ea 

7° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.00244 

.00244 

~0038T 

.00382 

.00548 

.00551 

.00745 

.00751 

0 

1 

.00246 

.00246 

.00383 

.00385 

.00551 

.00554 

.00749 

.00755 

1 

2 

.00248 

.00248 

.00386 

.00387 

.00554 

.00557 

.00752 

.00758 

2 

3 

.00250 

.00250 

.00388 

.00390 

.00557 

J00560 

.00756 

.00762 

3 

4 

.00252 

.00252 

.00391 

.00392 

.00560 

.00563 

.00760 

.00765 

4 

5 

.00254 

.00254 

.00393 

.00395 

.00563 

.00566 

.00763 

.00769 

5 

6 

.00256 

.00257 

.00396 

.00397 

.00566 

.00569 

.00767 

.00778 

6 

7 

.00258 

.00259 

.00398 

.00400 

.00569 

.00573 

.00770 

.00776 

7 

8 

.00260 

.00261 

.00401 

.00403 

.00572 

.00576 

.00774 

.00780 

8 

9 

.00262 

.00263 

.00404 

.00405 

.00576 

.00579 

.00778 

.00784 

9 

10 

.00264 

.00265 

.00406 

.00408 

.00579 

.00582 

.00781 

.00787 

10 

11 

.00266 

.00267 

.00409 

.00411 

.00582 

.00585 

.00785 

.00791 

11 

12 

.00269 

.00269 

.00412 

.00413 

.00585 

.00588 

.00789 

.00795 

12 

13 

.00271 

.00271 

.00414 

.00416 

.00588 

.00592 

.00792 

.00799 

13 

14 

.00273 

.00274 

.00417 

.00419 

.00591 

.00595 

.00796 

.00802 

14 

15 

.00275 

.00276 

.00420 

.00421 

.00594 

..00598 

.00800 

.00806 

15 

16 

.00277 

.00278 

.00422 

.00424 

.00598 

.00601 

.00803 

.00810 

16 

17 

.00279 

.00280 

.00425 

.00427 

.00601 

.00604 

.00807 

.00813 

17 

18 

.00281 

.00282 

.00428 

.00429 

.00604 

.00608 

.00811 

.00817 

18 

19 

.00284 

.00284 

.00430 

.00432 

.00607 

.00611 

.00814 

.00821 

19 

20 

.00286 

.00287 

.00433 

.00435 

.00610 

.00614 

.00818 

.00825 

20 

21 

.00288 

.00289 

.00436 

.00438 

.00614 

.00617 

.00822 

.00828 

21 

22 

.00290 

.00291 

.00438 

.00440 

.00617 

.00621 

.00825 

.00832 

22 

23 

.00293 

.00293 

.00441 

.00443 

.00620 

.00624 

.00829 

.00836 

23 

24 

.00295 

.00296 

.00444 

.00446 

.00623 

.OOG27 

.00833 

.00840 

24 

25 

.00297 

.00298 

.00447 

.00449 

.00626 

.00630 

.00837 

.00844 

25 

26 

.00299 

.00300 

.00449 

.00451 

.00630 

.C0634 

.C0840 

.00848 

26 

27 

.00301 

.00302 

.00452 

.00454 

.00633 

.00637 

.00844 

.00851 

27 

28 

.00304 

.00305 

.00455 

.00457 

.00636 

.00640 

.00848 

.00855 

28 

29 

.00306 

.00307 

.00458 

.00460 

.00640 

.00644 

.00852 

.00859 

29 

80 

.00308 

.00309 

.00460 

.00463 

.00643 

.00647 

.00856 

.00863 

30 

31 

.00311 

.00312 

.00463 

.00465 

.00646 

.00650 

.00859 

.00867 

31 

82 

.00313 

.00314 

.00466 

.00468 

.00649 

.00654 

.00863 

.00871 

32 

33 

.00315 

.00316 

.00469 

.00471 

.00653 

.00657 

.00867 

.00875 

33 

84 

.00317 

.00313 

.00472 

.00474 

.00656 

.00660 

.00871 

.00878 

34 

35 

.00320 

.00321 

.00474 

.00477 

.00659 

.00664 

.00875 

.00882 

35 

36 

.00322 

.00323 

.00477 

.00480 

.00663 

.00667 

.00878 

.00886 

36 

37 

.00324 

.00326 

.00480 

.00482 

.00666 

.00671 

.00882 

.00890 

37 

38 

.00327 

.00328 

.00483 

.00485 

.00669 

.00674 

.00886 

.00894 

38 

39 

'  .00329 

.00330 

.00486 

.00488 

.00673 

.00677 

.00890 

.00898 

39 

40 

.00332 

.00338 

.00489 

.00491 

.00676 

.00681 

.00894 

.00902 

40 

41 

.00334 

.00335 

.00492 

.00494 

.00680 

.00684 

.00898 

.00906 

41 

42 

.00336 

.00337 

.00494 

.00497 

.00683 

.00688 

.00902 

.00910 

42 

43 

.00339 

.00340 

.00497 

.00500 

.00686 

.00691 

.00906 

.00914 

43 

44 

.00341 

.00342 

.00500 

.00503 

.00690 

.00695 

.00909 

.00918 

44 

45 

.00343 

.00345 

.00503 

.00506 

.00693 

.00698 

.00913 

.00922 

45 

46 

.00346 

.00347 

.00506 

.00509 

.00697 

.00701 

.00917 

.00926 

46 

47 

.00348 

.00350 

.00509 

.00512 

.00700 

.00705 

.00921 

.00930 

47 

48 

.00351 

.00352 

.00512 

.00515 

.00703 

.00708 

.00925 

.00934 

48 

49 

.00353 

.00354 

.00515 

.00518 

.00707 

.00712 

.00929 

.00938 

49 

50 

.00356 

.00357 

.00518 

.00521 

.00710 

.00715 

.00933 

.00942 

50 

51 

.00358 

.00359 

.00521 

.00524 

.00714 

.00719 

.00937 

.00946 

51 

52 

.00361 

.00362 

.00524 

.00527 

.00717 

.00722 

.00941 

.00950 

52 

53 

.00363 

.00364 

.00527 

.00530 

.00721 

.00726 

.00945 

.00954 

53 

54 

.00365 

.00367 

.00530 

.00533 

.00724 

.00730 

.00949 

.00958 

54 

55 

.00368 

.00369 

.00533 

.00536 

.00728 

.00733 

.00953 

.00962 

55 

56 

.00370 

.00372 

.00536 

.00539 

.00731 

.00737 

.00957 

.00966 

56 

57 

.00373 

.00374 

.00539 

.00542 

.00735 

.00740 

.00961 

.00970 

57 

58 

.00375 

.00377 

.00542 

.00545 

.00738 

.00744 

.00965 

.00975 

58 

59 

.00378 

.00379 

.00545 

.00548 

.00742 

.00747 

.00969 

.00979 

59 

60 

.00381 

.00382 

.00548 

.00551 

.00745 

.00751  1 

.00973   .00983 

60 

NATURAL   VERSED  SINES  AND  EXTERNAL    SECANTS. 


8° 

9° 

10° 

11° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers.  ] 

Ex.  sec. 

0 

.00973 

.00983 

.01231 

.01247 

.01519 

.01543 

.01837 

.01872 

0 

1 

.00977 

.00987 

.01236 

.01251 

.01524 

.01548 

.01843 

.01877 

1 

2 

.00981 

.00991 

.01240 

.01256 

.01529 

.01553 

.01&48 

.01883 

2 

3 

.00985 

.00995  ' 

.01245 

.01261 

.01534 

.01558 

.01854 

.01889 

3 

4 

.00989 

.00999 

.01249 

.01265 

.01540 

.01564 

.01860 

.01895 

4 

5 

.00994 

.01004 

.01254 

.01270 

.01545 

.01569 

.01865 

.01901 

5 

6 

.00998 

.01008 

.01259 

.01275 

.01550 

.01574 

.01871 

.01906 

6 

7 

.01002 

.01012 

.01263 

.01279 

.01555 

.01579 

.01876 

.01912 

7 

8 

.01006 

.01016 

.01268 

.01284 

.01560 

.01585 

.01882 

.01918 

8 

9 

.01010 

.01020 

.01272 

.01289 

.01565 

.01590 

.01888 

.01924 

9 

10 

.01014 

.01024 

.01277 

.01294 

.01570 

.01595 

.01893 

.01930 

10 

11 

.01018 

.01029 

.01282 

.01298 

.01575 

.01601 

.01899 

.01936 

11 

12 

.01022 

.01033 

.01286 

.01303 

.01580 

.01606 

.01904 

.01941 

12 

13 

.01027 

.01037 

.01291 

.01308 

.01586 

.01611 

.01910 

.01947 

13 

14 

.01031 

.01041 

.01296 

.01313 

.01591 

.01616 

.01916 

.01953 

14 

15 

.01035 

.01046 

.01300 

.01318 

.01596 

.01622 

.01921 

.01959 

15 

16 

.01039 

.01050 

.01305 

.01322 

.01601 

.01627 

.01927 

.01965 

16 

17 

.01043 

.01054 

.01310 

.01327 

.01606 

.01633 

.01933 

.01971 

17 

18 

.01047 

.01059 

.01314 

.01332 

.01612 

.01638 

.01939 

.01977 

18 

19 

.01052 

.01063 

.01319 

.01337 

.01617 

.01643 

.01944 

.01983 

19 

20 

.01056 

.01067 

.01324 

.01342 

.01622 

.01649 

.01950 

.01989 

20 

21 

.01060 

.01071 

.01329 

.01346 

.01627 

.01654 

.01956 

.01995 

21 

22 

.01064 

.01076 

.01333 

.01351 

.01632 

.01659 

.01961 

.02001 

22 

23 

.01069 

.01080 

.01338 

.01356 

.01638 

.01665 

.01967 

.02007 

23 

24 

.01073 

.01084 

.01343 

.01361 

.01643 

.01670 

.01973 

.02013 

24 

25 

.01077 

.01089 

.01348 

.01366 

.01648 

.01676 

.01979 

.02019 

25 

26 

.01081 

.01093 

.01352 

.01371 

.01653 

.01681 

.01984 

.02025 

26 

27 

.01086 

.01097 

.01357 

.01376 

.01659 

.01687 

.01990 

.02031 

27 

28 

.01090 

.01102 

.01362 

.01381 

.01664 

.01692 

.01996 

.02037 

28 

29 

.01094 

.01106 

.01307 

.01386 

.01669 

.01698 

.02002 

.02043 

29 

30 

.01098 

.01111 

.01371 

.01391 

.01675 

.01703 

.02008 

.02049 

30 

31 

.01103 

.01115 

.01376 

.01395 

.01680 

.01709 

.02013 

.02055 

31 

32 

.01107 

.01119 

.01381 

.01400 

.01685 

.01714 

.02019 

.02061 

32 

33 

.01111 

.01124 

.01386 

.01405 

.01690 

.01720 

.02025 

.02067 

33 

34 

.01116 

.01128 

.01391 

.01410 

.01696 

.01725 

.02031 

.02073 

34 

35 

.01120 

.01133 

.01396 

.01415 

.01701 

.01731 

.02037 

.02079 

35 

36 

.01124 

.01137 

.01400 

.01420 

.01706 

.01736 

.02042 

.02085 

36 

37 

.01129 

.01142 

.01405 

.01425 

.01712 

.01742 

.02048 

.02091 

37 

38 

.01133 

.01146 

.01410 

.01430 

.01717 

.01747 

.02054 

.02097 

38 

39 

.01137 

.01151 

.01415 

.01435 

.01723 

.01753 

.020GO 

.02103 

39 

40 

.01142 

.01155 

.01420 

.01440 

.01728 

.01758 

.02066 

.02110 

40 

41 

.01146 

.01160 

.01425 

.01445 

.01733 

.01764 

.02072 

.02116 

41 

42 

.01151 

.01164 

.01430 

.01450 

.01739 

.01769 

.02078 

.02122 

42 

43 

.01155 

,01169 

.01435 

.01455 

.01744 

.01775 

.02084 

.02128 

43 

44 

.01159 

.01173 

.01439 

.01461 

.01750 

.01781 

.02090 

.02134 

44 

45 

.01164 

.01178 

.01444 

.01466 

.01755 

.01786 

.02095 

.02140 

45 

46 

.01168 

.01182 

.01449 

.01471 

.01760 

.01792 

.02101 

.02146 

46 

47 

.01173 

.01187 

.01454 

•01476 

.01766 

.01798 

.02107 

.02153 

47 

48 

.01177 

.01191 

.01459 

.01481 

.01771 

.01803 

.02113 

.02159 

48 

49 

.01182 

.01196 

.01464 

.01486 

.01777 

.01809 

.02119 

,02165 

49 

50 

.01186 

.01200 

.01469 

.01491 

.01782 

.01815 

.02125 

.02171 

50 

51 

.01191 

.01205 

.01474 

.01496 

.01788 

.01820 

.02131 

.02178 

51 

52 

.01195 

.01209 

.01479 

.01501 

.01793 

.01826 

.02137 

.02184 

52 

53 

.01200 

.01214 

.01484 

.01506 

.01799 

.01832 

.02143 

.02190 

53 

54 

.01204 

.01219 

.01489 

.01512 

.01804 

.01837 

.02149 

.02196 

54 

55 

.01209 

.01223 

.01494 

.01517 

.01810 

.01843 

.02155 

.02203 

55 

56 

.01213 

.01228 

.01499 

.01522 

.01815 

.01849 

.02161 

.02209 

56 

57 

.01218 

.01233 

.01504 

.01527 

.01821 

.01854 

.02167 

.02215 

57 

58 

.01222 

.01237 

.01509 

.01532 

.01826 

.01860 

.02173 

.02221 

58 

59 

.01227 

.01242 

.01514 

.01537 

.01832 

.01866 

.02179 

.02228 

59 

60 

.01231 

.01247 

.01519 

.01543 

.01837 

.01872 

.02185 

.02234 

60 

NATURAL  VERSED  SINES  AND  EXTERNAL  SECANTS, 


f 

12° 

13° 

14° 

15° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.02185 

.02234 

.02563 

.02630 

.02970 

.03061 

.03407 

.03528 

0 

1 

.02191 

.02240 

.02570 

.02637 

.02977 

.03069 

.03415 

.03536 

1 

2 

.02197 

.02247 

.02576 

.02644 

.02985 

.03076  i 

.03422 

.03544 

2 

3 

.02203 

.02253 

.025&3 

.02651 

.02992 

.03084 

.03430 

.03552 

3 

4 

.02210 

.02259 

.02589 

.02658 

.02999 

.03091  ! 

.03438 

.03560 

4 

5 

.02216 

.02266 

.02596 

.02665 

.03006 

.03099  j 

.03445 

.03568 

5 

6 

.02222 

.02272 

.02602 

.02672 

.03013 

.03106 

.03453 

.03576 

6 

7 

.02228 

.02279 

.02609 

.02679 

.03020 

.03114 

.03460 

.03584 

7 

8 

.02234 

.02285 

.02616 

.C2686 

.03027 

.03121 

.03468 

.03592 

8 

9 

.02240 

.02291 

.02622 

.02693 

.03034 

.03129 

.03476 

.03601 

9 

10 

.02246 

.02298 

.02629 

.02700 

.03041 

.03137 

.03483 

.03609 

10 

11 

.02252 

.02304 

.02635 

.02707 

.03048 

.03144 

.03491 

.03617 

11 

12 

.02258 

.02311 

.02642 

.02714 

.03055 

.03152 

.03498 

.03625 

12 

13 

.02265 

.02317 

.02649 

.02721 

.03063 

.03159 

.03506 

.03633 

13 

14 

.02271 

.02323 

.02655 

.02728 

.03070 

.03167 

.03514 

.03642 

14 

15 

.02277 

.02330 

.02662 

.02735 

.03077 

.03175 

.03521 

.03650 

15 

16 

.02283 

.02336 

.02669 

.02742 

.03084 

.03182 

.03529 

.03658 

16 

17 

.02289 

.02343 

.02675 

.02749 

.03091 

.03190 

.03537 

.03666 

17 

18 

.02295 

.02349 

.02682 

.02756 

.03098 

.03198 

.03544 

.03674 

18 

19 

.02302 

.02356 

.02689 

.02763 

.03106 

.03205 

.03552 

.03683 

19 

20 

.02308 

.02362 

.02696 

.02770 

.03113 

.03213 

.03560 

.03691 

20 

21 

.02314 

.02369 

.02702 

.02777 

.03120 

.03221 

.03567 

.03699 

21 

22 

.02320 

.02375 

.02709 

.02784 

.03127 

.03228 

.03575 

.03708 

22 

23 

.02327 

.02382 

.02716 

.02791 

.03134 

.03236 

.03583 

.03716 

23 

24 

.02333 

.02388 

.02722 

.02799 

.03142 

.03244 

.03590 

.03724 

24 

25 

.02339 

.02395 

.02729 

.02806 

.03149 

.03251 

.03598 

.03732 

25 

26 

.02345 

.02402 

.02736 

.02813 

.03156 

.03259 

.03006 

.03741 

26 

27 

.02352 

.02408 

.02743 

.02820 

.03163 

.03267 

.03614 

.03749 

27 

28 

.02358 

.02415 

.02749 

.02827 

.03171 

.03275 

.03621 

.03758 

28 

29 

.02364 

.02421 

.02756 

.02834 

.03178 

.03282 

.03629 

.03766 

29 

30 

.02370 

.02428 

.02763 

.02842 

.03185 

.03290 

.03637 

.03774 

30 

31 

.02377 

.02435 

.02770 

.02849 

.03193 

.03298 

.03645 

.03783 

31 

32 

.02383 

.02441 

.02777 

.02856 

.03200 

.03306 

.03653 

.03791 

32 

33 

.02389 

.02448 

.02783 

.02863 

.03207 

.03313 

.03660 

.03799 

33 

34 

.02396 

.02454 

.02790 

.02870 

.03214 

.03321 

.03668 

.03808 

34 

35 

.02402 

.02461 

.02797 

.02878 

.03222 

.03329 

.03676 

.03816 

35 

36 

.02408 

.02468 

.02804 

.02885 

.03229 

.03337 

.03684 

.03825 

36 

37 

.02415 

.02474 

.02811 

.02892 

.03236 

.03345 

.03692 

.03833 

37 

38 

.02421 

.02481 

.02818 

.02899 

.03244 

.03353 

.03699 

.03842 

38 

39 

.02427 

.02488 

.02824 

.02907 

.03251 

.03360 

.03707 

.03850 

89 

40 

.02434 

.02494 

.02831 

.02914 

.03258 

.03368 

.03715 

.03858 

40 

41 

.02440 

.02501 

.02838 

.02921 

.03266 

.03376 

.03723 

.03867 

41 

42 

.02447 

.02508 

.02845 

.02928 

.03273 

.03384 

.03731 

.03875 

42 

43 

.02453 

.02515 

.02852 

.02936 

.03281 

.03392 

.03739 

.03884 

43 

44 

.02459 

.02521 

.02859 

.02943 

.03288 

.03400 

.03747 

.03892 

44 

45 

.02466 

.02528 

.02866 

.02950 

.03295 

.03408 

.03754 

.03901 

45 

46 

.02472 

.02535 

.02873 

.02958 

.03303 

.03416 

.03762 

.03909 

46 

47 

.02479 

.02542 

.02880 

.02965 

.03310 

.03424 

.03770 

.03918 

47 

48 

.02485 

.02548 

.02887 

.02972 

.03318 

.03432 

.03778 

.03927 

48 

49 

.02492 

.02555 

.02894 

.02980 

.03325 

.03439 

.03786 

.03935 

49 

50 

.02498 

.02562 

.02900 

.02987 

.03333 

.03447 

.03794 

.03944 

50 

51 

.02504 

.02569 

.02907 

.02994 

.03340 

.03455 

.03802 

.03952 

51 

52 

.02511 

.02576 

.02914 

.03002 

.03347 

.03463 

.03810 

.03961 

52 

53 

.02517 

.02582 

.02921 

.03009 

.03355 

.03471 

.03818 

.03969 

53 

54 

.02584 

.02589 

.02928 

.03017 

!  .03362 

.03479 

.03826 

.03978 

54 

55 

.02530 

.02596 

.02935 

.03024 

.03370 

.03487 

.03834 

.03987 

55 

56 

.02537 

.02603 

.02942 

.03032 

.03377 

.03495 

.03842 

.03995 

56 

57 

.02543 

.02610 

.02949 

.03039 

.03385 

.03503 

.03850 

.04004 

57 

58 

.02550 

.02617 

.02956 

.03046 

.03392 

.03512 

.03858 

.04013 

58 

59 

.02556 

.02624 

.02963 

.03054 

.03400 

.03520 

.03866 

.04021 

59 

60 

.02563 

.02630 

.02970 

.03061 

.03407 

.03528 

.03874 

.04030 

60 

NATURAL  VERSED  SINES  AND  EXTERNAL  SECANTS. 


/ 

16* 

lta           18* 

19° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex,  sec. 

Vers. 

Ex.  sec. 

0 

.03874 

.04030   .04370 

.04569 

.04894 

.05146 

.05448 

.057'62 

0 

1 

.03882 

.04039 

.04378 

.04578 

.04903 

.05156 

.05458 

.05773 

1 

2 

.03890 

.04047 

.04387 

.04588 

.04912 

.05166 

.05467 

.05783 

2 

3 

.03898 

.04056 

.04395 

.04597 

.04921 

.05176 

.05477 

.05794 

3 

4 

.03906 

.04065 

.04404 

.04606 

.04930 

.05186 

.05486 

.05805 

4 

5 

.03914 

.04073 

.04412 

.04616 

.04939 

.05196 

.05496 

.05815 

5 

6 

.03922 

.04082 

.04421 

.04625 

.04948 

.05206 

.05505 

.05826 

6 

7 

.03930 

.04091 

.04429 

.04635 

.04957 

.05216 

.05515 

.05836 

7 

8 

.03938 

.04100 

.04438 

.04644 

.04967 

.05226 

.05524 

.05847 

8 

9 

.03946 

.04108 

.04446 

.04653 

.04976 

.05236 

.05534 

.05858 

9 

10 

.03954 

.04117 

.04455 

.04663 

.04985 

.05246 

.05543 

.05869 

10 

11 

.03963 

.04126 

.04464 

.04672 

.04994 

.05256 

.05553 

.05879 

11 

12 

.03971 

.04135  i 

.04472 

.04682 

.05003 

.05266 

.05562 

.05890 

12 

13 

.03979 

.04144  ! 

.04481 

.04691 

.05012 

.05276 

.05572 

.05901 

13 

14 

.03987 

.04152  I 

.04489 

.04700 

.05021 

.05286 

.05582 

.05911 

14 

15 

.03995 

.04161 

.04498 

.04710 

.05030 

.05297  i 

.05591 

.05922 

15 

16 

.04003 

.04170 

.04507 

.04719 

.05039 

.05307  I 

.05601 

.05933 

16 

17 

.04011 

.04179 

.04515 

.04729 

.05048 

.05317  i 

.05610 

.05944 

17 

18 

.04019 

.04188 

.04524 

.04738 

.05057 

.05327 

.05620 

.05955 

18 

19 

.04028 

.04197 

.04533 

.04748 

.05067 

.C5337 

.05630 

.05965 

19 

20 

.04036 

.04206 

.04541 

.04757 

.05076 

.05347 

.05639 

.05976 

20 

21 

.04044 

.04214 

.04550 

.04767 

.05085 

.05357 

.05649 

.05987 

21 

22 

.04052 

.04223 

.04559 

.04776 

.05094 

.C5367 

.05658 

.05998 

22 

23 

.04060 

.04232 

.04567 

.04786 

.05103 

.05378 

.05668 

.06009 

23 

24   .04069 

.04241 

.04576 

.04795 

.05112 

.05388  ! 

.05678 

.06020 

24 

25 

.04077 

.04250 

.04585 

.04805 

.05122 

.C5398 

.05687 

.06030 

25 

26 

.04085 

.04259 

.04593 

.04815 

.05131 

.C5408 

.05697 

.06041 

26 

27 

.04093 

.04268 

.04602 

.04824 

.05140 

.05418 

.05707 

.06052 

27 

28 

.04102 

.04277 

.04611 

.04834 

.C5149 

.05429 

.05716 

.06063 

28 

29 

.04110 

.04286 

.04620 

.04843 

.05158 

.C5439 

.05726 

.06074 

29 

30 

.04118 

.04295 

.04628 

.04853 

.05168 

.05449 

.05736 

.06085 

30 

31 

.04126 

.04304 

.04637 

.04863 

.05177 

.05460 

.05746 

.06096 

31 

32 

.04135 

.04313 

.04646 

.04872 

.C5186 

.C5470 

.05755 

.06107 

32 

33 

.04143 

.04322 

.04655 

.04882 

.05195 

.05480 

.05765 

.06118 

33 

34 

.04151 

.04331 

.04663 

.04891 

.05205 

.05490 

.05775 

.06129 

34 

35 

.04159 

.04340 

.04672 

.04901 

.05214 

.C5501 

.05785 

.06140 

35 

36 

.04168 

.04349 

.04681 

.04911 

.05223 

.05511 

.05794 

.06151 

36 

37 

.04176 

.04358 

.04690 

.04920 

.05232 

.05521 

.05804 

.06162 

37 

38 

.04184 

.04367 

.04699 

.04930 

.05242 

.05532 

.05814 

.06173 

38 

39 

.04193 

.04376 

.04707 

.04940 

.05251 

.C5542 

.05824 

.06184 

39 

40 

.04201 

.04385 

.04716 

.04950 

.05260 

.05552 

.05833 

.06195 

40 

41 

.04209 

.04394 

.04725 

.04959 

.05270 

.05563 

.05843 

.06206 

41 

42 

.04218 

.04403 

.04734 

.04969 

.C5279 

.05573 

.05853 

.06217 

42 

43 

.04226 

.04413 

.04743 

.04979 

.05288 

.05584 

.05863 

.06228 

43 

44 

.04234 

.04422 

.04752 

.04989 

.05298 

.05594 

.05873 

.06239 

44 

45 

.04243 

.04431 

.04760 

.04998 

.05307 

.05604 

.05882 

.06250 

45 

46 

.04251 

.04440 

.04769 

.05008 

.05316 

.05615  ! 

.05892 

.06261 

46 

47 

.04260 

.04449 

.04778 

.05018 

.05326 

.05625 

.05902 

.06272 

47 

48 

.04268 

.04458 

.04787 

.05028 

.05335 

.05636 

.05912 

.06283 

48 

49 

.04276 

.04468 

.04796 

.05038 

.05344 

.05646 

.05922 

.06295 

49 

50 

.04285 

.04477 

.04805 

.05047 

.05354 

.05657  | 

.05932 

.06306 

50 

51 

.04293 

.04486 

.04814 

.05057 

.05363 

.05667 

.05942 

.06317 

51 

52 

.04302 

.04495 

.04823 

.05067 

.05373 

.05678 

.05951 

.06328 

52 

53 

.04310 

.04504 

.04832 

.05077 

.05382 

.05688 

.05961 

.06339 

53 

54 

.04319 

.04514 

.04841 

.05087 

.05391 

.05699 

.05971 

.06350 

54 

55 

.04327 

.04523 

.04850 

.05097 

.05401 

.05709 

.05981 

.06362 

55 

56 

.04336 

.04532 

.04858 

.05107 

.05410 

.05720 

.05991 

.06373 

56 

57 

.04344 

.04541 

.04867 

.05116 

.05420 

.05730 

.06001 

.06384 

57 

58 

.04353 

.04551 

.04876 

.05126 

.05429 

.05741 

.06011 

.06895 

58 

59 

.04361 

.04560 

.04885 

.05136 

.05439 

.05751 

.06021 

.0&407 

59 

60 

.04370 

.04569 

.04894 

.05146 

.05448 

.05762 

.06031 

.06418 

60 

91 


NATURAL  VERSED    SINES  AND  EXTERNAL    SECANTS, 


/ 

20° 

21' 

22° 

23° 

i 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec.  i 

Vers. 

Ex.  sec. 

Vers.  Ex.  sec. 

~T 

.06031 

.06418 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

0 

i 

.06041 

.06429 

.06652 

.07126 

.07293 

.07866 

.07961 

.08649 

1 

2 

.06051 

.06440 

.06663 

.07138 

.07303 

.07879 

.07972 

.08663 

2 

3 

.06061 

.06452 

.06673 

.07150 

.07314 

.07892 

.07984 

.08676 

3 

4 

.06071 

.06463 

.06684 

.07162 

.07325 

.07904 

.07995 

.08690 

4 

5 

.06081 

.06474 

.06694 

.07174 

.07336 

.07917 

.08006 

.08703 

5 

6 

.06091 

.06486 

.06705 

.07186 

.07347 

.07930 

.08018 

.08717 

6 

7 

.06101 

.06497 

.06715 

.07199 

.07358 

.07943 

.08029 

.08730 

7 

8 

.06111 

.06508 

.06726 

.07211 

.07369 

.07955 

.08041 

.08744 

8 

9 

.06121 

.06520 

.06736 

.07223 

.07380 

.07968 

.08052 

.08757 

9 

10 

.06131 

.06531 

.06747 

.07235 

.07391 

.07981 

.08064 

.08771 

10 

11 

.06141 

.06542 

.06757 

.07247 

.07402 

.07994 

.08075 

.08784 

11 

12 

.06151 

.06554 

.06768 

.07259 

.07413 

.08006 

.08086 

.08798 

12 

13 

.06161 

.06565 

.06778 

.07271 

.07424 

.08019 

.08098 

.08811 

13 

14 

.06171 

.06577 

.06789 

.07283 

.07435 

.08032 

.08109 

.08825 

14 

15 

.06181 

.06588 

.00799 

.07295 

.07446 

.08045 

.08121 

.08839 

15 

16 

.06191 

.06600 

.06810 

.07307 

.07457 

.08058 

.08132 

.08852 

16 

IT 

.06201 

.06611 

.06820 

.07320 

.07468 

.08071 

.08144 

.08866 

17 

18 

.06211 

.06622 

.06831 

.07332 

.07479 

.08084 

.08155 

.08880 

18 

19 

.06221 

.06634 

.06841 

.07344 

.07490 

.08097 

.08167 

.08893 

19 

20 

.06231 

.06645 

.06852 

.07356 

.07501 

.08109 

.08178 

.08907 

20 

21 

.06241 

.06657 

.06863 

.07368 

.07512 

.08122 

.08190 

.08921 

21 

22 

.06252 

.06668 

.06873 

.07380 

.07523 

.08135 

.08201 

.08934 

22 

23 

.06262 

.06680 

.06884 

.07393 

.07534 

.08148 

.08213 

.08948 

23 

24 

.06272 

.06691 

.06894 

.07405 

.07545 

.08161 

.08225 

.08962 

24 

25 

.06282 

.06703 

.06905 

.07417 

.07556 

.08174 

.08236 

.08975 

25 

26 

.06292 

.06715 

.06916 

.07429 

.07568 

.08187 

.08248 

.08989 

26 

27 

.06302 

.06726 

.06926 

.07442 

.07579 

.08200 

.08259 

.09003 

27 

28 

.06312 

.06738 

.06937 

.07454 

.07590 

.08213 

.08271 

.09017 

28 

29 

.06323 

.06749 

.06948 

.07466 

.07601 

.08226 

.08282 

.09030 

20 

30 

.06333 

.06761 

.06958 

.07479 

.07612 

.08239 

.08294 

.09044 

30 

31 

.06343 

.06773 

.06969 

.07491 

.07623 

.08252 

.08306 

.09058 

31 

32 

.06353 

.06784 

.06980 

.07503 

.07634 

.08265 

.08317 

.09072 

32 

33 

.06363 

.'06796 

.06990 

.07516 

.07645 

.08278 

.08329 

.09086 

33 

34 

.06374 

.06807 

.07001 

.07528 

.07657 

.08291 

.08340 

.09099 

34 

35 

.06384 

.06819 

.07012 

.07540 

.07668 

.08305 

.08352 

.09113 

35 

36 

.06394 

.06831 

.07022 

.07553 

.07679 

.08318 

.08364 

.09127 

36 

37 

.06404 

.06843 

.07033 

.07565 

.07690 

.08331 

.08375 

.09141 

37 

38 

.06415 

.06854 

.07044 

.07578 

.07701 

.08344 

.08387 

.09155 

33 

39 

.06425 

.06866 

.07055 

.07590 

.07713 

.08357 

.08399 

.09169 

39 

40 

.06435 

.06878 

.07065 

.07602 

.07724 

.08370 

.08410 

.09183 

40 

41 

.06445 

.06889 

.07076 

.07615 

.07735 

.08383 

.08422 

.09197 

41 

42 

.06456 

.06901 

.07087 

.07627 

.07746 

.08397 

.08434 

.09211 

42 

43 

.06466 

.06913 

.07098 

.07640 

.07757 

.08410 

.08445 

.09224 

43 

44 

.06476 

.06925 

.07108 

.07652 

.07769 

.08423 

.08457 

.09238 

44 

45 

.06486 

.06936 

.07119 

.07665 

.07780 

.08436 

.08469 

.09252 

45 

46 

.06497 

.06948 

.07130 

.07677 

.07791 

.08449 

.08481 

.09266 

46 

47 

.06507 

.06960 

.07141 

.07690 

.07802 

.08463 

.08492 

.09280 

47 

48 

.06517 

.06972 

.07151 

.07702 

.07814 

.08476 

.08504 

.09294 

43 

49 

.06528 

.06984 

.07162 

.07715 

.07825 

.08489 

.08516 

.09308 

40 

50 

.06538 

.06995 

.07173 

.07727 

.07836 

.08503 

.08528 

.09323 

50 

51 

.06548 

.07007 

.07184 

.07740 

.07848 

.08516 

.08539 

.09a37 

51 

52 

.06559 

.07019 

.07195 

.07752 

.07859 

.08529 

.08551 

.09351 

52 

53 

.06569 

.07031 

.07206 

.07765 

.07870 

.08542 

.08563 

.09365 

53 

54 

.06580 

.07043 

.07216 

.07778 

.07881 

.08556 

.08575 

.09379 

54 

55 

.06590 

.07055 

.07227 

.07790 

.07893 

.08569 

.08586 

.09393 

55 

56 

.06600 

.07067 

.07238 

.07803 

.07904 

.08582 

.08598 

.09407 

56 

57 

.06611 

.07079 

.07249 

.07816 

.07915 

.08596 

.08610 

.09421 

57 

58 

.06621 

.07091 

.07260 

.07828 

.07927 

.08609 

.08622 

.09435 

58 

59 

.06632 

.07103 

.07271 

.07841 

.07938 

.08623 

.08634 

.09449 

59 

60 

.06648 

.07115 

.07282 

.07853 

.07950 

.08636 

.08645 

.09464 

60 

•NATURAL  VERSED    SINES    AND  EXTERNAL   SECANTS. 


/ 

24° 

25° 

26° 

27° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  cec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.08645 

.09464 

.09369 

.10338 

.10121 

.11260 

.10899 

.12233 

0 

1 

.08657 

.09478 

.09382 

.10353 

.10133 

.11276 

.10913 

.12249 

1 

2 

.08669 

.09492 

.09394 

.10368 

.10146 

.11292 

.10926 

.12266 

2 

3 

.08681 

.09506 

.09406 

.10383 

.10159 

.11308 

.10939 

.12283 

3 

4 

.08693 

.09520 

.09418 

.10398 

.10172 

.11323 

.10952 

.12299 

4 

5   .08705 

.09535 

.09431 

.10413 

.10184 

.11339 

.10965 

.12316 

5 

G   .08717 

.09549 

.09443 

.10428 

.10197 

.11355 

.10979 

.12333 

6 

7   .08728 

.09563 

.09455 

.10443 

.10210 

.11371 

.10992 

.12349 

7 

8 

.08740 

.09577 

.09468 

.10458 

.10223 

.11387 

.110C5 

.12366 

8 

9 

.08752 

.09592 

.09480 

.10473 

.10236 

.11403 

.11019 

.12383 

9 

10 

.08764 

.09606 

.09493 

.10488 

.10248 

.11419 

.11032 

.12400 

10 

11 

.08776 

.09620 

.09505 

.10503 

.10261 

.11435 

.11045 

.12416 

11 

12 

.08788 

.09635 

.09517 

.10518 

.10274 

.11451 

.11058 

.12433 

12 

13 

.08800 

.09649 

.C9530 

.10533 

.10287 

.11467 

.11072 

.12450 

13 

14 

.08812 

.09663 

.09542 

.10549 

.10300 

.11483 

.11085 

.12467 

11 

15 

.08824 

.09678 

.09554 

.10564 

.10313 

.11499 

.11098 

.12484 

15 

16 

.08836 

.09692 

.09567 

.10579 

.10326 

.11515 

.11112 

.12501 

1G 

17 

.08848 

.09707 

.09579 

.10594 

.10328 

.11531 

.11125 

.12518 

17 

18 

.08860 

.09721 

.09592 

.10609 

.10351 

.11547 

.11138 

.12534 

13 

19 

.08872 

.09735 

.09604 

.10625 

.103G4 

.11563 

.11152 

.12551 

19 

20 

.08884 

.09750 

.09617 

.10640 

.10377 

.11579 

.11165 

.12568 

20 

21 

.08896 

.09764 

.09629 

.10655 

.10390 

.11595 

.11178 

.12585 

21 

22 

.08903 

.09779 

.C9G42 

.10670 

.10403 

.11611 

.11192 

.12602 

22 

23 

.08920 

.09793 

.09654 

.10686 

.10416 

.11627 

.11205 

.12619 

23 

24 

.08932 

.09808 

.09666 

.10701 

.10429 

.11643 

.11218 

.12636 

24 

25 

.08944 

.09822 

.09679 

.10716 

.10442 

.11659 

.11232 

.12653 

25 

26 

.08956 

.09837 

.09691 

.10731 

.10455 

.11675 

.11245 

.12670 

26 

27 

.08968 

.00851 

.09704 

.10747 

.10468 

.11691 

.11259 

.12687 

27 

28 

.08980 

.09866 

.09716 

.10762 

.10481 

.11708 

.11272 

.12704 

28 

29 

.08992 

.09880 

.C9729 

.10777 

.10494 

.11724 

.11285 

.12721 

29 

30 

.09004 

.09895 

.09741 

.10793 

.10507 

.11740 

.11299 

.12738 

30 

31 

.09016 

.09909 

.09754 

.10808 

.10520 

.11756 

.11312 

.12755 

31 

32 

.09028 

.09924 

.09707 

.10824 

.10533 

.11772 

.11326 

.12772 

32 

33 

.09040 

.09930 

.08779 

.10839 

.10546 

.11789 

.11339 

.12789 

33 

34 

.09052 

.09953 

.09792 

.10854 

.10559 

.11805 

.11353 

.12807 

34 

85 

.09064 

.09908 

.09804 

.10870  :;  .10572 

.11821 

.11366 

.12824 

35 

£6 

.09076 

.09982 

.09817 

.10885 

.10585 

.11838 

.11380 

.12841 

3G 

37 

.09089 

.09997 

.09829 

.10901 

.10598 

.11854 

.11393 

.12858 

37 

88 

.09101 

.10012 

.09842 

.10916 

.10611 

.11870 

.11407 

.12875 

38 

39 

.09113 

.10026 

.09854 

.10932 

.10624 

.11886 

.11420 

.12892 

30 

40 

.09125 

.10041 

.09867 

.10947 

.10637 

.11903 

.11434 

.12910 

40 

(1 

.09137 

.10053 

.09880 

.10963 

.10650 

.11919 

.11447 

.12927 

41 

42 

.09149 

.10071 

.C9892 

.10978 

.106G3 

.11936 

.11461 

.12944 

42 

43 

.09161 

.10085 

.09905 

.10994 

.10676 

.11952 

.11474 

.12961 

43 

44 

.09174 

.10100 

.09918 

.11009 

.10689 

.11968 

.11488 

.12979 

44 

45 

.09186 

.10115 

.09930 

.11025 

.10702 

.11985 

.11501 

.12996 

45 

46 

.09198 

.10130 

.09943 

.11041 

.10715 

.12001 

.11515 

.13013 

46 

47 

.09210 

.10144 

.09955 

.11056 

.10728 

.12018 

.11528 

.13031 

47 

48 

.09222 

.10159 

.09968 

.11072 

.10741 

.12034 

.11542 

.13048 

48 

49 

.09234 

.10174 

.09981 

.11087 

.10755 

.1.2051 

.11555 

.13065 

49 

50 

.09247 

.10189 

.09993 

.11103 

.10768 

.12067 

.11569 

.13083 

50 

51 

.09259 

.10204 

.10006 

.11119 

.10781 

.12084 

.11583 

.13100 

51 

52 

.09271 

.10218 

.10019 

.11134 

.10794 

.12100 

.11596 

.13117 

52 

53 

.09283 

.10233 

.10032 

.11150 

.10807 

.12117 

.11610 

.13135 

53 

54 

.09296 

.10248 

.10044 

.11166 

.10820 

.12133 

.11623 

.13152 

54 

55 

.09308 

.10263 

.10057 

.11181 

.10833 

.12150 

.11637 

.13170 

55 

56 

.09320 

.10278 

.10070 

.11197 

.10847 

.12166 

.11651 

.13187 

56 

57 

.09332 

.  10293 

.10082 

.11213 

.10860 

.12183 

.11664 

.13205 

57 

58 

.09345 

.10308 

.10095 

.11229 

.10873 

.12199 

.11678 

.13222 

58 

59 

.09357 

.10323 

.10108 

.11244 

.10886 

.12216 

.11692 

.13240 

59 

60 

.09369 

.  10338 

.  10121 

.11260 

.10899 

.12233 

.11705 

.13257 

60 

NATURAL    VERSED    SINES    AND  EXTERNAL  SECANTS. 


' 

28° 

29° 

30° 

31° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.11705 

.13257 

.12538 

.14335 

.13307 

.15470 

.14283 

.16663 

0 

i 

.11719 

.13275 

.12552 

.14354 

.13412 

.15489 

.14298 

.16684 

1 

2 

.11733 

.13292 

.12566 

.14372 

.13427 

.15509 

.14313 

.16704 

2 

ft 

.11746 

.13310 

.12580 

.14391 

.13441 

.15528 

.14328 

.16725 

3 

4 

.11700 

.13327 

.12595 

.14409 

.13456 

.15548 

.14343 

.16745 

4 

5 

.11774 

.13345 

.12609 

.14428 

.1347'0 

.15567 

.14358 

.16766 

5 

C 

.11787 

.13362 

.12623 

.14446 

.13485 

.15587 

.14373 

.16786 

6 

7 

.11801 

.13380 

.12637 

.14465 

.13499 

.15006 

.14388 

.16806 

7 

8 

.11815 

.13398 

.12651 

.14483 

.13514 

.15626 

.14403 

.16827 

8 

9 

.11823 

.13415 

.12665 

.14502 

.13529 

.15045 

.14418 

.16848 

9 

10 

.11842 

.  13433 

.12679 

.14521 

.13543 

.15665 

.14433 

.16868 

10 

11 

.11850 

.13451 

.12694 

.14539 

.13558 

.15684 

.14449 

.16889 

11 

12 

.11670 

.13408 

.12708 

.14558 

.13573 

.15704 

.11404 

.16909 

12 

13 

.11883 

.13486 

.12722 

.14576 

.13587 

.15724 

.14479 

.16930 

13 

14 

.11897 

.13504 

.12736 

.14595 

.13602 

.15743 

.1-1404 

.16950 

14 

15 

.11911 

.13521 

.12750 

.14614 

.13616 

.15763 

.11509 

.16971 

15 

16 

.11925 

.13539 

.12765 

.14632 

.13631 

.15782 

.14524 

.16992 

16 

17 

.11933 

.13557 

.12779 

.14651 

.13646 

.15802 

.11539 

.17012 

17  1 

18 

.11952 

.13575 

.127103 

.14670 

.13660 

.15822 

.14554 

.17033 

18 

19 

.11366 

.13593 

.12807 

.14689 

.13075 

.15841 

.14569 

.17C54 

19 

20 

.11980 

.13610 

.12822 

.14707 

.13690 

.15861 

.14584 

.17075 

20 

21 

.11994 

.13628 

.12836 

.14726 

.13705 

.15881 

.14599 

.17095 

21 

22 

.12007 

.13646 

.12850 

.11745 

.13719 

.15901 

.14615 

.17116 

22 

23 

.12021 

.13664 

.12864 

.14704 

.13734 

.15920 

.14630 

.17137 

23 

21 

.12035 

.13682 

.12879 

.14782 

.13749 

.15940 

.14645 

.17158 

24 

25 

.12049 

.13700 

.12893 

.14801 

.13703 

.15960 

.14660 

.17178 

25 

28 

.12063 

.13718 

.12907 

.14820 

.13778 

.15980 

.14675 

.17199 

26 

27 

.12077 

.13735 

.12921 

.14839 

.13793 

.16000 

.14690 

.17220 

27 

28 

.12091 

.13753 

.12936 

.14858 

.13803 

.10019 

.14703 

.17241 

28 

29 

.12104 

.13771 

.12950 

.1-1077 

.13822 

.16039 

.14721 

.17262 

29 

£0 

.12118 

.13789 

.12964 

.14896 

.13837 

.10059 

.14736 

.17283 

30 

El 

.12132 

.13807 

.12979 

.14914 

.13852 

.16079 

.14751 

.17304 

31 

32 

.12146 

.13825 

.12993 

.14933 

.13807 

.10003 

.14703 

.17325 

32 

83 

.12160 

.13843 

.13007 

.14952 

.13881 

.10119 

.14782 

.17346 

33 

34 

.12174 

.13861 

.13022 

.14971 

.13893 

.10139 

.14797 

.17367 

34 

So 

.12188 

.13879 

.13036 

.14990 

.13911 

.101.9 

.14812 

.17388 

35 

36 

.12202 

.13897 

.13051 

.15009 

.13923 

.10179 

.14827 

.17409 

36 

37 

.12216 

.13916 

.13065 

.15028 

.13941 

.16199 

.14843 

.17430 

37 

38 

.12230 

.13934 

.13079 

.15047 

.13955 

.16219 

.14853 

.17451 

38 

£9 

.12244 

.13952 

.13094 

.15066 

.13970 

.162:-  9 

.14873 

.  17472 

39 

40 

.12257 

.13970 

.13108 

.15085 

.13985 

.10259 

.14888 

.17493 

40 

41 

.12271 

.13983 

.13122 

.15105 

.14000 

.16279 

.14904 

.17514 

41 

42 

.12285 

.14006 

.13137 

.15124 

.14015 

.16299 

.14919 

.17535 

42 

43 

.12299 

.14024 

.13151 

.15143 

.14030 

.16319 

.14904 

.17556 

43 

44 

.12313 

.14042 

.13166 

.15162 

.14044 

.16339 

.14949 

.17577 

44 

45 

.12327 

.14061 

.13180 

.15181 

.14059 

.16359 

.14905 

.17598 

45 

46 

.12341 

.14079 

.13195 

.15200 

.14074 

.16380 

.14980 

.17620 

4(5 

47 

.12355 

.14097 

.13209 

.15219 

.14089 

.16400 

.14905 

.17641 

47 

48 

.12369 

.14115 

.13223 

.15239 

.14104 

.16420 

.15011 

.17662 

43 

49 

.12383 

.14134 

.13238 

.15258 

.14119 

.10440 

.15023 

.17683 

49 

50 

.12397 

.14152 

.13252 

.15277 

.14134 

.16460 

.15041 

.17704 

50 

51 

.12411 

.14170 

.13267 

.15*96 

.14149 

.16481 

.15057 

.17726 

51 

52 

.12425 

.14188 

.13281 

.15315 

.14164 

.16501 

.15072 

.17747 

52 

53 

.12439 

.14207 

.13296 

.15335 

.14179 

.16521 

.15087 

.17768 

53 

54 

.12454 

.14225 

.13310 

.15354 

.  14194 

.16541 

.15103 

.17790 

54 

55 

.12468 

.14243 

.13325 

.15373 

.14208 

.16562 

.15118 

.17811 

55 

56 

.12482 

.14262 

.13339 

.15393 

.14223 

.16582 

.15134 

.17832 

56 

57 

.12496 

.14280 

.13354 

.15412 

.14238 

.16602 

.15149 

.17854 

57 

58 

.12510 

.14299 

.13368 

.15431 

.14253 

.16623 

.15164 

.17875 

58 

59 

.12524 

.14317 

.13383 

.15451 

.14268 

.16643 

.15180 

.17896 

59 

60 

.12538 

.14335 

.13397 

.15470 

.14283 

.  16663 

.15195 

.17918 

60 

NATURAL  VERSED    SINES   AND   EXTERNAL   SECANTS. 


/ 

32" 

33° 

34° 

35° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.15195 

.17918 

.16133 

.19236 

.17096 

.20622 

.18085 

.22077 

0 

1 

.15211 

.17939 

.16149 

.19259 

.17113 

.20645 

.18101 

.22102 

1 

2 

.  15226 

.17961 

.16165 

.19281 

.17129 

.20669 

.18118 

.22127 

2 

3 

.15241 

.17982 

.16181 

.19304 

.17145 

.20693 

.18135 

.22152 

3 

4 

.15257 

.18004 

.16196 

.19327 

.17161 

.20717 

.18152 

.22177 

4 

5 

.15272 

.18025 

.16212 

.19349 

.17178 

.20740 

.18168 

.22202 

5 

6 

.15288 

.  18047 

.16228 

.19372 

.17194 

.20764 

.18185 

.22227 

6 

7 

.15303 

.18068 

.16244 

.19394 

.17210 

.20788 

.18202 

.22252 

r 

8 

.15319 

.18090 

.16260 

.19417 

.17227 

.20812 

.18218 

.22277 

8 

9 

.15334 

.18111 

.16276 

.19440 

.17243 

.20836 

.18235 

.22302 

9 

10 

.15350 

.18133 

.16292 

.19463 

.17259 

.20859 

.18252 

.22327 

10 

11 

.15365 

.18155 

.16308 

.194&5 

.17276 

.20883 

.18269 

.22352 

11 

12 

.15381 

.18176 

.16324 

.19508 

.17292 

.20907 

.18286 

.22377 

13 

13 

.15396 

.18198 

.16340 

.19531 

.17308 

.20931 

.18302 

.22402 

13 

14 

.15412 

.18220 

.16355 

.19554 

.17325 

.20955 

.18319 

.22428 

14 

15 

.15427 

.18241 

.16371 

.19576 

.17341 

.20979 

.18336 

.22453 

15 

16 

.15443 

.18263 

.16387 

.19599 

.17357 

.21003 

.18353 

.22478 

16 

17 

.15458 

.18285 

.16403 

.19622 

.17374 

.21027 

.18369 

.22503 

17 

18 

.15474 

.18307 

.16419 

.19645 

.17390 

.21051 

.18386 

.22528 

18 

19 

.15489 

.18328 

.16435 

.19668 

.17407 

.21075 

.18403 

.22554 

19 

20 

.15505 

.18350 

.16451 

.19691 

.17423 

.21099 

.18420 

.22579 

20 

21 

.15520 

.18372 

.16467 

.19713 

.17439 

.21123 

.18437 

.22604 

21 

22 

.15536 

.18394 

.16483 

.19736 

.17453 

.21147 

.18454 

.22629 

22 

23 

.15552 

.18416 

.16499 

.19759 

.17472 

.21171 

.18470 

.22655 

23 

24 

.15567 

.18437 

.16515 

.19782 

.17483 

.21195 

.18487 

.22680 

24 

25 

.15583 

.18459 

.16531 

.19805 

.17505 

.21220 

.18504 

.22706 

25 

26 

.15598 

.18481 

.16547 

.19828 

.17522 

.21244 

.18521 

.22731 

26 

27 

.15614 

.18503 

.16563 

.19851 

.17533 

.21268 

.18538 

.22756 

27 

28 

.15630 

.18525 

.16579 

.19874 

.17554 

.21292 

.18555 

.22782 

28 

29 

.  15645 

.18547 

.16595 

.19897 

.17571 

.21316 

.18572 

.22807 

29 

30 

.15661 

.18569 

.16611 

.19920 

.17587 

.21341 

.18588 

.22833 

30 

31 

.15676 

.18591 

.16627 

.19944 

.17604 

.21365 

.18605 

.22858 

31 

32 

.15692 

.18613 

.16644 

.19967 

.17620 

.21389 

.18622 

.22884 

32 

33 

.15708 

.18635 

.16660 

.19990 

.17637 

.21414 

.18639 

.22909 

33 

34 

.15723 

.18657 

.16676 

.20013 

.17653 

.21438 

.18656 

.22935 

34 

35 

.15780 

.18679 

.16692 

.20036 

.17670 

.21462 

.18673 

.22960 

35 

36 

.15755 

.18701 

.16708 

.20059 

.17686 

.21487 

.18690 

.22986 

36 

37 

.15770 

.18723 

.16724 

.20083 

.17703 

.21511 

.18707 

.23012 

37 

38 

.15786 

.18745 

.16740 

.20106 

.17719 

.21535 

.18724 

.23037 

38 

39 

.15803 

.18767 

.16756 

.20129 

.17736 

.21560 

.18741 

.23063 

39 

40 

.15818 

.18790 

.16772 

.20152 

.17752 

.21584 

.18758 

.23089 

40 

41 

.15833 

.18812 

.16788 

.20176 

.17769 

.21609 

.18775 

.23114 

41 

42 

.15849 

.18834 

.16805 

.20199 

.17786 

.21633 

.18792 

.23140 

42 

43 

.15865 

.18856 

.16821 

.20222 

.17802 

.21658 

.18809 

.23166 

43 

44 

.15880 

.18878 

.16837 

.20246 

.17819 

.21682 

.18826 

.23192 

44 

45 

.15896 

.18901 

.16853 

.20269 

.17835 

.21707 

.18843 

.23217 

45 

46 

.15912 

.18923 

.16869 

.20292 

.17852 

.21731 

.18860 

.23243 

46 

47 

,15923 

.18945 

.16885 

.20316 

.17868 

.21756 

.18877 

.23269 

47 

48 

.15943 

.18967 

.16902 

.20339 

.17885 

.21781 

.18894 

.23295 

48 

49 

.15959 

.18990 

.16918 

.20363 

.17902 

.21805 

.18911 

.23321 

49 

50 

.15975 

.19012 

.16934 

.20386 

.17918 

.21830 

.18928 

.23347 

50 

51 

.15991 

.19034 

.16950 

.20410 

.17935 

.21855 

.18945 

.23373 

51 

52 

.16006 

.19057 

.16966 

.20433 

.17952 

.21879 

.18962 

.23399 

52 

53 

.16022 

.19079 

.16983 

.20457 

.17968 

.21904 

.18979 

.23424 

53 

54 

.16038 

.19102 

.16999 

.20480 

.17985 

.21929 

.18996 

.23450 

54 

55 

.16054 

.19124 

.17015 

.20504 

.18001 

.21953 

.19013 

.23476 

55 

56 

.16070 

.19146 

.17031 

.20527 

.18018 

.21978 

.19030 

.23502 

56 

57 

.16085 

.19169 

.  17047 

.20551 

.18035 

.22003 

.19047 

.23529 

57 

58 

.16101 

.19191 

.17064 

.20575 

.18051 

.22028 

.19064 

.23555 

58 

59 

.16117 

.19214 

.17080 

.20598 

.18068 

.22053 

.19081 

.23581 

59 

60 

.16133 

.19236  i 

.17096 

.20622 

.18085 

.22077 

.19098 

.23607 

60 

NATURAL  VERSED   SINES   AND   EXTERNAL    SECANTS. 


/ 

36° 

37° 

38° 

39° 

t 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.19098 

.23607 

.20136 

.25214 

.21199 

.26902 

.22285 

.28676 

0 

1 

.19115 

.23633 

.20154 

.25241 

.21217 

.26931 

.22304 

.287'06 

1 

2 

.19133 

.23659 

.20171 

.25269 

.21235 

.26960 

.22322 

.28737 

2 

3 

.19150 

.23685 

.20189 

.25296 

.21253 

.26988 

.22340 

.28767 

3 

4 

.19167 

.23711 

.20207 

.25324 

.21271 

.27017 

.22359 

.28797 

4 

5 

.19184 

.23738 

.20224 

.25351 

.21289 

.27046 

.22377 

.28828 

5 

6 

.19201 

.23764 

.20242 

.25379 

.21307 

.27075 

.22395 

.28858 

6 

7 

.19218 

.23790 

.20259 

.25406 

.21324 

.27104 

.22414 

.28889 

7 

8 

.19235 

.23816 

.20277 

.25434 

.21342 

.27133 

.22432 

.28919 

8 

9 

.19252 

.23843 

.20294 

.25462 

.21360 

.27162 

.22450 

.28950 

9 

10 

.19270 

.23869 

.20312 

.25489 

.21378 

.27191 

.22469 

.28980 

10 

11 

.19287 

.23895 

.20329 

.25517 

.21396 

.27221 

.22487 

.29011 

11 

12 

.19304 

.23922 

.20347 

.25545 

.21414 

.27250 

.2250G 

.29042 

12 

13 

.19321 

.23948 

.20305 

.25572 

.21432 

.27279 

.22524 

.29072 

13 

14 

.19338 

.23975 

.20382 

.25600 

.21450 

.27308 

.22542 

.29103 

14 

15 

.19356 

.24001 

.20400 

.25628 

.21468 

.27337 

.22561 

.29133 

15 

16 

.19373 

.24028 

.20417 

.25656 

.21486 

.27366 

.22579 

.29164 

16 

17 

.19390 

.24054 

.20435 

.25683 

.21504 

.27396 

.22598 

.29195 

17 

38 

.19407 

.24081 

.20453 

.25711 

.21522 

.27425 

.22616 

.29226 

18 

19 

.19424 

.24107 

.20470 

.25739 

.21540 

.27454 

.22634 

.29256 

19 

£0 

.19442 

.24134 

.20488 

.25767 

.21558 

.27483 

.22653 

.29287 

20 

21 

.19459 

.241  GO 

.20506 

.25795 

.21576 

.27513 

.22671 

.29318 

21 

22 

.19476 

.24187 

.20523 

.25823 

.21505 

.27542 

.22690 

.29349 

22 

23 

.19493 

.24213 

.20541 

.25851 

.21613 

.27572 

.22708 

.29380 

23 

24 

.19511 

.24240 

.20559 

.25879 

.21631 

.27601 

.22727 

.29411 

24 

25 

.19528 

.24267 

.20576 

.25907 

.21649 

.27630 

.22745 

.29442 

25 

26 

.19545 

.24293 

.20594 

.25935 

.21667 

.27660 

.22764 

.29473 

26 

27 

.  19562 

.24320 

.20612 

.25963 

,21685 

.27689 

.22782 

.29504 

27 

28 

.19580 

.24347 

.20629 

.25991 

.21703 

.27719 

.22801 

.29535 

28 

29 

.19597 

.24373 

.20647 

.26019 

.21721 

.27748 

.22819 

.29566 

29 

30 

.19614 

.24400 

.20665 

.26047 

.21739 

.27778 

.22838 

.29597 

30 

31 

.19632 

.24427 

.20682 

.26075 

.21757 

.27807 

.22856 

.29628 

31 

32 

.19649 

.24454 

.20700 

.26104 

.21775 

.27837 

.2287'5 

.29659 

32 

33 

.19666 

.24481 

.20718 

.26132 

.21794 

.27867 

.22893 

.29690 

33 

34 

.1^684 

.24508 

.20736 

.26160 

.21812 

.27896 

.22912 

.29721 

34 

35 

.19?'01 

.24534 

.20753 

.26188 

.21830 

.27926 

.22930 

.29752 

35 

36 

.19718 

.24561 

.20771 

.26216 

.21848 

.27956 

.22949 

.29784 

36 

37 

.19736 

.24588 

.20789 

.26245 

.21866 

.27985 

.22907 

.29815 

37 

38 

.19753 

.24615 

.20807 

.26273 

.21884 

.28015 

.22986 

.29846 

38 

39 

.19770 

.24642 

.20824 

.26301 

.21902 

.28045 

.23004 

.29877 

39 

40 

.19788 

.24669 

.20842 

.26330 

.21921 

.28075 

.23023 

.29909 

40 

41 

.19805 

.24696 

.20860 

.26358 

.21939 

.28105 

.23041 

.29940 

41 

42 

.19822 

.24723 

.20878 

.26387 

.21957 

.28134 

.23060 

.29971 

42 

43 

.19840 

.24750 

.20895 

.26415 

.21975 

.28164 

.23079 

.30003 

43 

44 

.19857 

.24777 

.20913 

.26443 

.21993 

.28194 

.23097 

.30034 

44 

45 

.1987'5 

.24804 

.20931 

.26472 

.22012 

.28224 

.23116 

.30066 

45 

46 

.19892 

.24832 

.20949 

.26500 

.22030 

.28254 

.23134 

.30097 

46 

47 

.19909 

.24859 

.20967 

.26529 

.22048 

.28284 

.23153 

.30129 

47 

48 

.19927 

.24886 

.20985 

.26557 

.22066 

.28314 

.23172 

.30160 

48 

49 

.19944 

.24913 

.21002 

.26586 

.22084 

.28344 

.23190 

.30192 

49 

50 

.19962 

.24940 

.21020 

.26615 

.22103 

.28374 

.23209 

.30223 

50 

51 

.19979 

.24967 

.21038 

.26643 

.22121 

.28404 

.23228 

.30255 

51 

52 

.19997 

.24995 

.21056 

.26672 

.22139 

.28434 

.23246 

.30287 

52 

53 

.20014 

.25022 

.21074 

.26701 

.22157 

.28464 

.23265 

.30318 

53 

54 

.20032 

.25049 

.21092 

.26729 

.22176 

.28495 

.23283 

.30350 

54 

55 

.20049 

.25077 

.21109 

.26758  ! 

.22194 

.28525 

.23302 

.30382 

55 

56 

.20066 

.25104 

.21127 

.26787  | 

.22212 

.28555 

.23321 

.30413 

56 

57 

.20084 

.25131 

.21145 

.26815 

.22231 

.28585 

.23339 

.30445 

57 

58 

.20101 

.25159 

.21163 

.26844 

.22249 

.28615 

.23358 

.30477 

58 

59 

.20119 

.25186 

.21181 

.26873 

.22267 

.28646 

.23377 

.30509 

59 

60 

.20136 

.25214 

.21199 

.26902 

.22285 

.28676 

.23396 

.30541 

60 

NATURAL  VERSED   SINES  AND   EXTERNAL   SECANTS. 


• 

40° 

41° 

42° 

43° 

/ 

Vers. 

Ex.  sec 

Vers. 

Ex.  sec. 

Vers. 

Ex,  sec. 

Vers. 

Ex.  sec. 

0 

.23396 

.30541 

.24529 

.32501 

.25686 

.34563 

.26865 

.36733 

0 

1 

.23414 

.30573 

.24548 

.32535 

.25705 

.34599 

.26884 

.36770 

1 

2 

.23-133 

.30605 

.24567 

.32568 

.25724 

.34634 

.26904 

.36807 

2 

3 

.23452 

.30636 

.24586 

.32602 

.25744 

.34669 

.26924 

.36844 

3 

4 

.23470 

.306G8 

.24605 

.32636 

.25763 

.34704 

.26944 

.36881 

4 

5 

.23489 

.30700 

.24625 

.32669 

.25783 

.34740 

.26964 

.36919 

5 

6 

.23508 

.30732 

.24644 

.32703 

.25802 

.34775 

.26984 

.36956 

6 

7 

.23527 

.30764 

.24663 

.32737 

.25822 

.34811 

.27004 

.36993 

7 

8 

.23545 

.30796 

.24682 

.32770 

.25841 

.34846 

.27024 

.37030 

8 

9 

.235G4 

'.30829 

.24701 

.32804 

.25861 

.34882 

.27043 

.37068 

9 

10 

.23583 

.30861 

.24720 

.32838 

.25880 

.34917 

.27063 

.37105 

10 

11 

.23602 

.30893 

.24739 

.32872 

.25900 

.34953 

.27083 

.37143 

11 

12 

.23620 

.30925 

24759 

.32905 

.25920 

.34988 

.271C3 

.37180 

12 

13 

.23639 

.30957 

.*24778 

.32939 

.259C9 

.35024 

.27123 

.37218 

13 

14 

.23658 

.30989 

.24797 

.32973 

.25959 

.35060 

.27143 

.37255 

14 

15 

.23677 

.31022 

.24816 

.33007 

.25978 

.35095 

.27163 

.37293 

15 

16 

.23696 

.31054 

.24835 

.33041 

.25993 

.35131 

.27183 

.37330 

16 

17 

.23714 

.31086 

.24854 

.33075 

.26017 

.35167 

.27203 

.37368 

17 

18 

.23733 

.31119 

.24874 

.33109 

.26037 

.35203 

.27223 

.37406 

18 

19 

.23752 

.31151 

.24803 

.33143 

.26056 

.35238 

.27243 

.37443 

19 

20 

.23771 

.31183 

.24912 

.33177 

.26076 

.35274 

.27263 

.37481 

20 

21 

.23790 

.31216 

.24931 

.33211 

.26096 

.35310 

.27283 

.37519 

21 

22 

.23808 

.31248 

.24950 

.33245 

.26115 

.35346 

.27303 

.37556 

22 

23 

.23827 

.31281 

.24970 

.33279 

.26135 

.35382 

.27323 

.37594 

23 

24 

.23846 

.31313 

.24989 

.33314 

.26154 

.35418 

.27343 

.37632 

24 

25 

.23865 

.31346 

.25008 

.33348 

.26174 

.35454 

.273C3 

.37670 

25 

26 

.23884 

.31378 

.25027 

.33382 

.26194 

.35490 

.27383 

.37708 

26 

27 

.23903 

'  .31411 

.25047 

.33416 

.26213 

.35526 

.27403 

.37746 

27 

28 

.23922 

.31443 

.25066 

.33451 

.26233 

.35562 

.27423 

.37784 

28 

29 

.23941 

.31476 

.25085 

.33485 

.26253 

.35598 

.27443 

.37822 

29 

30 

.23959 

.31509 

.25104 

.33519 

.26272 

.35634 

.27463 

.37860 

30 

31 

.23973 

.31541 

.25124 

.33554 

.26292 

.35670 

.27483 

.37898 

31 

32 

.23997 

.31574 

.25143 

.33588 

.26312 

.35707 

.27503 

.37936 

32 

33 

.24016 

.31607 

.251G2 

.33622 

.26331 

.35743 

.27523 

.37974 

33 

34 

.24035 

.31610 

.25182 

.33657 

.26351 

.35779 

.27543 

.38012 

34 

35 

.24054 

.31672 

.25201 

.33691 

.26371 

.35815 

.27563 

.38051 

35 

36 

.24073 

.31705 

.25220 

.33726 

.26390 

.35852 

.27583 

.38089 

36 

37 

.24092 

.31738 

.25240 

.33760 

.26410 

.35888 

1  .27603 

.38127 

37 

38 

.24111 

.31771 

.25259 

.33795 

.26430 

.35924 

.27623 

.38165 

33 

39 

.24130 

.31804 

.25278 

.33830 

.26449 

.35961 

.27643 

.38204 

39 

40 

.24149 

.31837 

.25297 

.33864 

.26469 

.35997 

.27663 

.38242 

40 

<!! 

.24168 

.31870 

.25317 

.33899 

.26489 

.36034 

.27683 

.38280 

41 

42 

.24187 

.31903 

.25336 

.33934 

.26509 

.36070 

.27703 

.38319 

42 

43 

.24206 

.31936 

.25356 

.33968 

.26528 

.36107 

.27723 

.38357 

43 

44 

.24225 

.31969 

.25375 

.34003 

.26548 

.36143 

.27743 

.38396 

44 

45 

.24244 

.32002 

.25394 

.34038 

.26568 

.36180 

.27764 

.38434 

45 

46 

.24262 

.32035 

.25414 

.3407'3 

.26588 

.36217 

.27784 

.38473 

46 

47 

.24281 

.32068 

.25433 

.34108 

.26607 

.36253 

.27804 

.38512 

47 

48 

.24300 

.32101 

.25452 

.34142 

.26627 

.36290 

.27824 

.38550 

48 

49 

.24320 

.32134 

.25472 

.34177 

.26647 

.36327 

.27844 

.38589 

49 

50 

.24339 

.32168 

.25491 

.34212 

.26667 

.36363 

.27864 

.38628 

50 

51 

.24358 

.32201 

.25511 

.34247 

.26686 

.36400 

.27884 

.38666 

51 

52 

.24377 

.32234 

.25530 

.34282 

.26706 

.36437 

.27905 

.38705 

52 

53 

.24396 

.32267 

.25549 

.34317 

.26726 

.36474 

.27925 

.38744 

53 

54 

.24415 

.32301 

.25569 

.34352 

.26746 

.36511 

.27945 

.38783 

54 

55 

.24434 

.32334 

.25588 

.34387 

.26766 

.36548 

.27965 

.38822 

55 

56' 

.24453 

.32368 

.25608 

.34423 

.26785 

.36585 

.27985 

.38860 

56 

57 

.2^472 

.32401 

.25627 

.34458 

.26805 

.36622 

.28005 

.38899 

57 

58 

.24491 

.32434 

.25647 

.34493 

.26825 

.36659 

.28026 

.38938 

58 

59 

.24510 

.32468 

.25666 

.34528 

.26845 

.36696 

.28046 

.38977 

59 

60 

.24529 

.32501 

.25686 

.34563 

.26865 

.36733 

.28066 

.39016 

60 

NATURAL,  VERSED   SINES  AND  EXTERNAL,  SECANTS. 


' 

44° 

45° 

46' 

47° 

' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.28066 

.39016 

.29289 

.41421 

.30534 

.43956 

.31800 

.46628 

0 

1 

.28086 

.39055 

.29310 

.41463 

.30555 

.43999 

.31821 

.46674 

1 

2 

.28106 

.39095 

.29330 

.41504 

.30576 

.44042 

.31843 

.46719 

2 

3 

.28127 

.39134 

.29351 

.41545 

.30597 

.44086 

.31864 

.46765 

3 

4 

.28147 

.39173 

.29372 

.41586 

.30618 

.44129 

.31885 

.46811 

4 

5 

.28167 

.39212 

.29392 

.41627 

.30639 

.44173 

.31907 

.46857 

5 

G 

.28187 

.39251 

.29413 

.41669 

.30660 

.44217 

.31928 

.46903 

6 

7 

.28208 

.39291 

.29433 

.41710 

.30681 

.44260 

.31949 

.46949 

7 

8 

.28228 

.39330 

.29454 

.41752 

.30702 

.44304 

.31971 

.46995 

8 

9 

.28248 

.39369 

.29475 

.41793 

.30723 

.44347 

.31992 

.47041 

9 

10 

.28268 

.39409 

.29495 

.41835 

.30744 

.44391 

.32013 

.47087 

10 

11 

.28289 

.39448 

.29516 

.41876 

.30765 

.44435 

.32035 

.47134 

11 

12 

.28309 

.39487 

.29537 

.41918 

.30786 

.44479 

.32056 

.47180 

12 

13 

.28329 

.39527 

.29557 

.41959 

.30807 

.44523 

.32077 

.47226 

13 

14 

.28350 

.39566 

.29578 

.42001 

.30828 

.44567 

.32099 

.47272 

14 

15 

.28370 

.39606 

.29599 

.42042 

.30849 

.44610 

.32120 

.47319 

15 

16 

.28390 

.39646 

.29619 

.42084 

.30870 

.44654 

.32141 

.47365 

16 

17 

.28410 

.39685 

.29640 

.42126 

.30891 

.44698 

.32163 

.47411 

17 

18 

.28431 

.39725 

.29661 

.42168 

.30912 

.44742 

.32184 

.47458 

18 

19 

.28451 

.39764 

.29681 

.42210 

.30933 

.44787 

.32205 

.47504 

19 

20 

.28471 

.39804 

.29702 

.42251 

.30954 

.44831 

.32227 

.47551 

20 

21 

.28492 

.39844 

.29723 

.42293 

.30975 

.44875 

.32248 

.47598 

21 

22 

.28512 

.39884 

.29743 

.42335 

.30906 

.44019 

.32270 

.47644 

22 

23 

.28532 

.39924 

.29764 

.42377 

.31017 

.44963 

.32291 

.47691 

23 

24 

.28553 

.39963 

.29785 

.42419 

.31038 

.45007 

.32312 

.47738 

24 

25 

.28573 

.40003 

.29805 

.42461 

.31059 

.45052 

.32334 

.4778-1 

25 

26 

.28593 

.40043 

.29826 

.42503 

.31080 

.45096 

.32355 

.47831 

26 

27 

.28614 

.40083 

.29847 

.42545 

.31101 

.45141 

.32377 

.47878 

27 

28 

.28634 

.40123 

.29868 

.42587 

.31122 

.45185 

.32398 

.47925 

28 

29 

.28655 

.40163 

.29888 

.42630 

.31143 

.45229 

.32420 

,47972 

29 

80 

.28675 

.40203 

.29909 

.42672 

.31105 

.45274 

.32441 

.48019 

30 

31 

.28695 

.40243 

.29930 

.42714 

.31186 

.45319 

.32462 

.48066 

31 

32 

.28716 

.40283 

.29951 

.42756 

.31207 

.45363 

.32484 

.48113 

32 

33 

.28736 

.40324 

.29971 

.42799 

.31223 

.45408 

.32505 

.48160 

33 

34 

.28757 

.40304 

.29992 

.42841 

.31249 

.45452 

.32527 

.48207 

34 

35 

.28777 

.40404 

.30013 

.42883 

.31270 

.45497 

.32548 

.48254 

35 

36 

.28797 

.40444 

.30034 

.42926 

.31291 

.45542 

.32570 

.48301 

36 

37 

.28818 

.40485 

.30054 

.42968 

.31312 

.45587 

.32591 

.48349 

37 

38 

.28838 

.40525 

.30075 

.43011 

.31334 

.45631 

.32613 

.48396 

38 

39 

.28859 

.40565 

.30096 

.43053 

.31355 

.45676 

.32634 

.48443 

39 

40 

.28879 

.40606 

.30117 

.43096 

.31376 

.45721 

.32656 

.48491 

40 

41 

.28900 

.40646 

.30138 

.43139 

.31397 

.45766 

.32677 

.48538 

41 

42 

.28920 

.40087 

.30158 

.43181 

.31418 

.45811 

.32699 

.48586 

42 

43 

.28941 

.40727 

.30179 

.43224 

.31439 

.45856 

.32720 

.48633 

43 

44 

.28961 

.40768 

.30200 

.43267 

.81461 

.45901 

.32742 

.48681 

44 

45 

.28981 

.40808 

.30221 

.43840 

.31482 

.45946 

.32763 

.48728 

45 

46 

.29002 

.40849 

.30242 

.43352 

.31503 

.45992 

.32785 

.48776 

46 

47 

.29022 

.40890 

.30263 

.43395 

.31524 

.46037 

.32806 

.48824 

47 

48 

.29043 

.40930 

.30283 

.43438 

.31545 

.46082 

.32828 

.48871 

48 

49 

.29063 

.40971 

.30304 

.43481 

.31567 

.46127 

.32849 

.48919 

49 

50 

.29084 

.41012 

.30325 

.43524 

.31588 

.46173 

.32871 

.48967 

50 

51 

.29104 

.41053 

.30346 

.43567 

.31609 

.46218 

.32893 

.49015 

51 

52 

.29125 

.41093 

.30367 

.43610 

.31630 

.46263 

.32914 

.49063 

52 

53 

.29145 

.41134 

.30388 

.43653 

.31651 

.46309 

.32936 

.49111 

53 

54 

.29166 

.41175 

.30409 

.43696 

.31673 

.46354 

.32957 

.49159 

54 

55 

.29187 

.41216 

.30430 

.43739 

.31694 

.46400 

.32979 

.49207 

55 

56 

.29207 

.41257 

.30451 

.43783 

.31715 

.46445 

.83001 

.49255' 

56 

57 

.29228 

.41298 

.30471 

.43826 

.31736 

.46491 

.33022 

.49303 

57 

58 

.29248 

.41339 

.30492 

.43869 

.31758 

.46537 

.33044 

.49351 

58 

59 

.29269 

.41380 

.30513 

.43912 

.31779 

.46582 

.33065 

.49399 

59 

60 

.29289 

.41421 

.30534 

.43956 

.31800 

.46628 

.33087 

.49448 

60 

98 


NATURAL  VERSED   SINES   AND  EXTERNAL   SECANTS. 


/ 

48* 

49° 

50- 

51* 

' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.33087 

.49448 

.34394 

.52425 

.35721 

.55572 

.37068 

.58902 

0 

1 

.33109 

.49496 

.34416 

.52476 

.35744 

.55626 

.37091 

.58959 

1 

2 

.33130 

.49544 

.34438 

.52527 

.35766 

.55680 

.37113 

.59016 

2 

3 

.33152 

.49593 

.34460 

.52579 

.35788 

.55734 

.37136 

.59073 

3 

4 

.33173 

.49641 

.34482 

.52630 

.35810 

.55789 

.37158 

.59130 

4 

5 

.33195 

.49690 

.34504 

.52681 

.35833 

.55843 

.37181 

.59188 

5 

6 

.33217 

.49738 

.34526 

.52732 

.35855 

.55897 

.37204 

.59245 

6 

7 

.33238 

.49787 

.34548 

.52784 

.35877 

.55951 

.37226 

.59302 

7 

8 

.33260 

.49835 

.34570 

.52835 

.35900 

.56005 

.37249 

.59360 

8 

9 

.33282 

.49884 

.34592 

.52886 

.35922 

.56060 

.37272 

.59418 

9 

10 

.33303 

.49933 

.34614 

.52938 

.35944 

.56114 

.37294 

.59475 

10 

11 

.33325 

.49981 

.34636 

.52989 

.35967 

.56169 

.37317 

.59533 

11 

12 

.33347 

.50030 

.34658 

.53041 

.35989 

.56223 

.37340 

.59590 

12 

13 

.33368 

.50079 

.34680 

.53092 

.38011 

.56278 

.37362 

.59648 

13 

14 

.33390 

.50128 

.34702 

.53144 

.36034 

.56332 

.37385 

.59706 

14 

15 

.33412 

.50177 

.34724 

.53196 

.36056 

.56387 

.37408 

.59764 

15 

16 

.33434 

.50226 

.34746 

.53247 

.36078 

.56442 

.37430 

.59822 

16 

17 

.33455 

.50275 

.34768 

.53299 

.36101 

.56497 

.37453 

.59880 

17 

18 

.33477 

.50324 

.34790 

.53351 

.36123 

.56551 

.37476 

.59938 

18 

19 

.33499 

.50373 

.34812 

.53403 

.36146 

.56606 

.37498 

.59996 

19 

20 

.33520 

.50422 

.34834 

.53455 

.36168 

.56661 

.37521 

.60054 

20 

21 

.33542 

.50471 

.34856 

.53507 

.36190 

.56716 

.37544 

.60112 

21 

22 

.33564 

.50521 

.34878 

.53559 

.36213 

.56771 

.37567 

.60171 

22 

23 

.33586 

.50570 

.34900 

.53611 

.36235 

.56826 

'.37589 

.60229 

23 

24 

.33607 

.50619 

.34923 

.53663 

.36258 

.56881 

.37612 

.60287 

24 

25 

.33629 

.50669 

.34945 

.53715 

.36280 

.56937 

.37635 

.60346 

25 

26 

.33651 

.50718 

.34967 

.53768 

.36302 

.56992 

.37658 

.60404 

26 

27 

.33673 

150767 

.34989 

.53820 

.36325 

.57047 

.37680 

.60463 

27 

28 

.33694 

.50817 

.35011 

.53872 

.36347 

.57103 

.37703 

.60521 

28 

29 

.33716 

.50866 

.35033 

.53924 

.36370 

.57158 

.37726 

.60580 

29 

30 

.33738 

.50916 

.35055 

.53977 

.36392 

.57213 

.37749 

.60639 

30 

31 

.33760 

.50966 

.35077 

.54029 

.36415 

.57269 

.37771 

.60698 

31 

32 

.33782 

.51015 

.35099 

.54082 

.36437 

.57324 

.37794 

.60756 

32 

33 

.33803 

.51065 

.35122 

.54134 

.36460 

.57380 

.37817 

.60815 

33 

34 

.33825 

.51115 

.35144 

.54187 

.36482 

.57436 

.37840' 

.60874 

34 

35 

.33847 

.51165 

.35166 

.54240 

.36504 

.57491 

.37'862 

.60933 

35 

36 

.33869 

.51215 

.35188 

.54292 

.36527 

.57547 

.37885 

.60992 

36 

37 

.33891 

.51265 

.35210 

.54345 

.36549 

.57603 

.37908 

.61051 

37 

38 

.33912 

.51314 

.35232 

.54398 

.36572 

.57659 

.37931 

.61111 

38 

39 

.33934 

.51364 

.35254 

.54451 

.36594 

.57715 

.37954 

.61170 

39 

40 

.33956 

.51415 

.35277 

.54504 

.36617 

.57771 

.37976 

.61229 

40 

41 

.33978 

.51465 

.35299 

.54557 

.36639 

.57827 

.37999 

.61288 

41 

42 

.34000 

.51515 

.35321 

.54010 

.36662 

.57883 

.38022 

.61348 

42 

43 

.34022 

.51565 

.35343 

.54663 

.36684 

.57939 

.38045 

.61407 

43 

44 

.34044 

.51615 

.33365 

.54716 

.36707 

.57995 

.38068 

.61467 

44 

45 

.34065 

.51665 

.35388 

.54769 

.36729 

.58051 

.38091 

.61526 

45 

46 

.34087 

.51716 

.35410 

.54822 

.36752 

.58108 

.38113 

.61586 

46 

47 

.34109 

.51766 

.35432 

.54876 

.36775 

.58164 

.38136 

.61646 

47 

48 

.34131 

.51817 

.35454 

.54929 

.36797 

.58221 

.38159 

.61705 

48 

49 

.34153 

.51867 

.35476 

.54982 

.36820 

.58277 

.38182 

.61765 

49 

50 

.34175 

.51918 

.35499 

.55036 

.36842 

.58333 

.38205 

.61825 

50 

51 

.34197 

.51968 

.35521 

.55089 

.36865 

.58390 

.38228 

.61885 

51 

52 

.34219 

.52019 

.35543 

.55143 

.36887 

.58447 

.38251 

.61945 

52 

53 

.34241 

.52069 

.35565 

.55196 

.36910 

.5850", 

.38274 

.62005 

53 

54 

.34262 

.52120 

.35588 

.55250 

.36932 

.585GO 

.38296 

.62065 

54 

55 

.34284 

.52171 

.35610 

.55303 

.36955 

.58617 

.38319 

.62125 

55 

56 

.34306 

.52222 

.35632 

.55357 

.36978 

.58674 

.38342 

.62185 

56 

57 

.34328 

.52273 

.35654 

.55411 

.37000 

.58731 

.38365 

.62246 

57 

58 

.34350 

.52323 

.35677 

.55465 

.37023 

.58788 

.38388 

.62306 

58 

59 

.34372 

.52374 

.35699 

.55518 

.37045 

.58845 

.38411 

.62366 

59 

60 

.34394 

.52425 

.35721 

.55572 

.37068 

.58902 

.38434 

.62427 

60 

99 


NATURAL   VERSED  SINES  AND  EXTERNAL  SECANTS, 


52° 

53° 

54« 

55° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.38434 

.62427 

.39819 

.66164 

.41221 

.70130 

.42642 

.74345 

0 

1 

.38457 

.62487 

.39842 

.66228 

.41245 

.70198 

.42666 

.74417 

1 

o 

.38480 

.62548 

.39865 

.66292 

.41269 

.70267 

.42690 

.74490 

2 

3 

.38503 

.62609 

.39888 

.66357 

.41292 

.70335 

.42714 

.74562 

3 

4 

.38526 

.62669 

.39911 

.66421 

.41316 

.70403 

.42738 

.74635 

4 

5 

.38549 

.62730 

.39935 

.66486 

.41339 

.70472 

.42762 

.74708 

5 

6 

.38571 

.62791 

.39958 

.66550 

.41363 

.70540 

.42785 

.74781 

6 

7 

.38594 

.62852 

.39981 

.66615 

.41386 

.70609 

.42809 

.74854 

7 

8 

.38617 

.62913 

.40005 

.66679 

.41410 

.70677 

.42833 

.74927 

8 

9 

.38640 

.62974 

.40028 

.66744 

.41433 

.70746 

.42857 

.75000 

9 

10 

.38663 

.63035 

.40051 

.66809 

.41457 

.70815 

.42881 

.75073 

10 

11 

.38686 

.63096 

.40074 

.66873 

.41481 

.70884 

.42905 

.75146 

11 

12 

.38709 

.63157 

.40098 

.66938 

.41504 

.70953 

.42929 

.75219 

12 

13 

.38732 

.63218 

.40121 

.67003 

.41528 

.71022 

.42953 

.75293 

13 

14 

.38755 

.63279 

.40144 

.67068 

.41551 

.71091 

.42976 

.75366 

14 

15 

.38778 

.63341 

.40168 

.67133 

.41575 

.71160 

.43000 

.75440 

15 

16 

.38801 

.63402 

.40191 

.67199 

.41599 

.71229 

.43024 

.75513 

16 

17 

.38824 

.63464 

.40214 

.67264 

.41622 

.71298 

.43048 

.75587 

17 

18 

.38847 

.63525 

.40237 

.67329 

.41646 

.71368 

.43072 

.75661 

18 

19 

.38870 

.6358? 

.40261 

.67394 

.41670 

.71437 

.43096 

.75734 

19 

20 

.38893 

.63648 

,40284 

.67460 

.41693 

.71506 

.43120 

.75808 

20 

21 

.38916 

.63710 

.40307 

.67525 

.41717 

.71576 

.43144 

.75882 

21 

22 

.38939 

.63772. 

.40331 

.67591 

.41740 

.71646 

.43168 

.75956 

22 

23 

.38962 

.63834 

.40354 

.67656 

.41764 

.71715 

.43192 

.76031 

23 

24 

.38985 

.63895 

.40378 

.67722 

.41788 

.71785 

.43216 

.76105 

24 

25 

.39009 

.63957 

.40401 

.67788 

.41811 

.71855 

.43240 

.76179 

25 

26 

.39032 

.64019 

.40424 

.67853 

.41835 

.71925 

.43264 

.76253 

26 

27 

.39055 

.64081 

.40448 

.67919 

.41859 

.71995 

.43287 

.76328 

27 

28 

.39078 

.64144 

.40471 

.67985 

.41882 

.72065 

.43311 

.76402 

28 

29 

.39101 

.64206 

.40494 

.68051 

.41906 

.72135 

.43335 

.76477 

29 

30 

.39124 

.64268 

.40518 

.68117 

.41930 

.72205 

.43359 

.76552 

30 

31 

.39147 

.64330 

.40541 

.68183 

.41953 

.72275 

.43383 

.76626 

31 

32 

.39170 

.64393 

.40565 

.68250 

.41977 

.72346 

.43407 

.76701 

32 

33 

.39193 

.64455 

.40588 

.68316 

.42001 

.72416 

.43431 

.76776 

33 

34 

.39216 

.64518 

.40611 

.'68382 

.42024 

.72487 

.43455 

.76851 

34 

35 

.39239 

.64580 

.40635 

.68449 

.42048 

.72557 

.43479 

.76926 

35 

36 

.39262 

.64643 

.40658 

.68515 

.42072 

.72628 

.43503 

.77001 

36 

37 

.39286 

.64705 

.40682 

.68582 

.42096 

.72698 

.43527 

.77077 

37 

38 

.39309 

.64768 

.40705 

.68648 

.42119 

.72769 

.43551 

.77152 

38 

39 

.39332 

.64831 

.40728 

.68715 

.42143 

.72840 

.43575 

.77227 

39 

40 

.39355 

.64894 

M0752 

.68782 

.42167 

.72911 

.43599 

.77303 

40 

41 

.39378 

.64957 

.40775 

.68848 

.42191 

.72982 

.43623 

.77378 

41 

42 

.39401 

.65020 

.40799 

.68915 

.42214 

.73053 

.43647 

.77454 

42 

43. 

.39424 

.65083 

.40822 

.68982 

.42238 

.73124 

.43671 

.77530 

43 

44 

.39447 

.65146 

.40846 

.69049 

.42262 

.73195 

.43695 

.77606 

44 

45 

.39471 

.65209 

.40869 

.69116 

.42285 

.73267 

.43720 

.77681 

45 

46 

.39494 

.65272 

.40893 

.69183 

.42309 

.73338 

.43744 

.77757 

46 

47 

.39517 

.65336 

.40916 

.69250 

.42333 

.73409 

.43768 

.77833 

47 

48 

.39540 

.65399 

.40939 

.69318 

.42357 

.73481 

.43792 

.77910 

48 

49 

.39563 

.65462 

.40963 

.69385 

.42381 

.73552 

.43816 

.77986 

49 

50 

.39586 

.65526 

.40986 

.69452 

.42404 

.73624 

.43840 

.78062 

50 

51 

.39610 

.65589 

.41010 

.69520 

.42428 

.73696 

.43864 

.78138 

51 

52 

.39633 

.65653 

.41033 

.69587 

.42452 

.73768 

.43888 

.78215 

52 

53 

.39656 

.65717 

.41057 

.69655 

.42476 

.73840 

.43912 

.78291 

53 

54 

.39679 

.65780 

.41080 

.69723 

.42499 

.73911 

.43936 

.78368 

54 

55 

.39702 

.65844 

.41104 

.69790 

.42523 

.73983 

.43960 

.78445 

55 

56 

.39726 

.65908 

.41127 

.69858 

.42547 

74056  ! 

.43984 

.78521 

56 

57 

.39749 

.65972 

.41151 

.69926 

.42571 

.74128  i 

.44008 

.78598 

57 

58 

.39772 

.66036 

.41174 

.69994 

.42595 

.74200  i 

.44032 

.78675 

68 

59 

.39795 

.66100 

.41198 

.70062 

.42619 

.74272 

.44057 

.78752 

59 

60 

.39819 

.66164 

.41221 

.70130 

.42642 

.74345 

.44081 

.78829 

60 

100 


NATURAL  VERSED   SINES  AND  EXTERNAL   SECANTS. 


t 

56° 

57° 

58« 

59" 

/ 

Vers. 

Ex.  sec 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.44081 

.78829 

.45536 

.83608 

.47008 

.88708 

.48496 

.94160 

0 

1 

.44105 

.78906 

.45560 

.83690 

.47033 

.88796 

.48521 

.94254 

1 

2 

.44129 

.78984 

.45585 

.83773 

.47057 

.88884 

.48546 

.94349 

2 

3 

.44153 

.79061 

.45609 

.83855 

.47082 

.88972 

.48571 

.94443 

3 

4 

.44177 

.79138 

.45634 

.83938 

.47107 

.89060 

.48596 

.94537 

4 

5 

.44201 

.79216 

.45658 

.84020 

.47131 

.89148 

.48621 

.94632 

5 

6 

.44225 

.79293 

.45683 

.84103 

.47156 

.89237 

.48646 

.94726 

6 

7 

.44250 

.79371 

.45707 

.84186 

.47181 

.89825 

.48671 

.94821 

7 

8 

.44274 

.79449 

.45731 

.84269 

.47206 

.89414 

.48696 

.94916 

8 

9 

.44298 

.79527 

.45756 

.84352 

.47230 

.89503 

.48721 

.95011 

9 

10 

.44322 

.79604 

.45780 

.84435 

.47255 

.89591 

.48746 

.95106 

10 

11 

.44346 

.79682 

.45805 

.84518 

.47280 

.89680 

.48771 

.95201 

11 

12 

.44370 

.79761 

.45829 

.84601 

.47304 

.89769 

.48796 

.95296 

12 

13 

.44395 

.79839 

.45854 

.84685 

.47329 

.89858 

.48821 

.95302 

13 

14 

.44419 

.79917 

.45878 

.84768 

.47354 

.89948 

.48846 

.95487 

14 

15 

.44443 

.79995 

.45903 

.84852 

.47379 

.90037 

.48871 

.95583 

15 

16 

.44467 

.80074 

.45927 

.84935 

.47403 

.90126 

.48896 

.95678 

16 

17 

.44491 

.80152 

.45951 

.85019 

.47428 

.90216 

.48921 

.95774 

17 

18 

.44516 

.80231 

.45976 

.85103 

.47453 

.90305 

.48946 

.95870 

18 

19 

.44540 

.80309 

.46000 

.85187 

.47478 

.90395 

.48971 

.95966 

19 

20 

.44504 

.80388 

.46025 

.85271 

.47502 

.90485 

.48996 

.96062 

20 

21 

.44588 

.80467 

.46049 

.85355 

.47527 

.90575 

.49021 

.96158 

21 

22 

.44612 

.80546 

.46074 

.85439 

.47552 

.90605 

.49046 

.96255 

22 

23 

.44637 

.80625 

.46098 

.85523 

.47577 

.90755 

.49071 

.96351 

23 

24 

.44661 

.80704 

.46123 

.85608 

.47601 

.90845 

.49096 

.96448 

24 

25 

.44685 

.80783 

.46147 

.85692 

.47626 

.90935 

.49121 

.96544 

25 

26 

.44709 

.  80862 

.46172 

.85777 

.47651 

.91026 

.49146 

.96641 

26 

27 

.44734 

.  $0942 

.46196 

.85861 

.47676 

.91116 

.49171 

.967'38 

27 

28 

.44758 

.81021 

.46221 

.85946 

.47701 

.91207 

.49196 

.96835 

28 

29 

.44782 

.81101 

.46246 

.86031 

.47725 

.91297 

.49221 

.96932 

29 

30 

.44806 

.81180 

.46270 

.86116 

.47750 

.91388 

.49246 

.97029 

30 

31 

.44831 

.81260 

.46295 

.86201 

.47775 

.91479 

.49271 

.97127 

31 

32 

.44855 

.81340 

.46319 

.86286 

.47800 

.91570 

.49296 

.97224 

32 

33 

.44879 

.81419 

.46344 

.86371 

.47825 

.91661 

.49321 

.97322 

33 

34 

.44903 

.81499 

.46368 

.86457 

.47&49 

.91752 

.49346 

.97420 

34 

35 

.44928 

.81579 

.46393 

.86542 

.47874 

.91844 

.49372 

.97517 

35 

36 

.44952 

.81659 

.46417 

.86627 

.47899 

.91935 

.49397 

.97615 

36 

37 

.44976 

.81740 

.46442 

.86713 

.47924 

.92027 

.49422 

.97713 

37 

38 

.45001 

.81820 

.46466 

.86799 

.47949 

.92118 

.49447 

.97811 

38 

39 

.45025 

.81900 

.46491 

.86885 

.47974 

.92210 

.49472 

.97910 

39 

40 

.45049 

.81981 

.46516 

.86990 

.47998 

.92302 

.49497 

.98008 

40 

41 

.45073 

.82061 

.46540 

.87056 

.48023 

.92394 

.49522 

.98107 

41 

42 

.45098 

.82142 

.46565 

.87142 

.48048 

.92486 

.49547 

.98205 

42 

43 

.45122 

.82222 

.46589 

.87229 

.48073 

.9.2578 

.49572 

.98304 

43 

44 

.45146 

.82303 

.46614 

.87315 

.48098 

.92670 

.49597 

.98403 

44 

45 

.45171 

.82384 

.46639 

.87401 

.48123 

.92762 

.49623 

.98502 

45 

46 

.45195 

.82465 

.46663 

.87488 

.48148 

.92855 

.49648 

.98601 

46 

47 

.45219 

.82546 

.46688 

.87574 

.48172 

.92947 

.49673 

.98700 

47 

48 

.45244 

.82627 

.46712 

.87661 

.48197 

.93040 

.49698 

.98799 

48 

49 

.45268 

.82709 

.46737 

.87748 

.48222 

.93133 

.49723 

.98899' 

49 

50 

.45292 

.82790 

.46762 

.87834 

.48247 

.93226 

.49748 

.98998 

50 

51 

.45317 

.82871 

.46786 

.87921 

.48272 

.93319 

.49773 

.99098 

51 

52 

.45341 

.82953 

.46811 

.88008 

.48297 

.93412 

.49799 

.99198 

52 

53 

.45365 

.83034 

.46836 

.88095 

.48322 

.93505 

.49824 

.99298 

53 

54 

.45390 

.83116 

.46860 

.88183 

.48347 

.93598 

.49849 

.99398 

54 

55 

.45414 

.831^8 

.46885 

.88270 

.48372 

.93692 

.49874 

.99498 

55 

56 

.45439 

.83*80 

.46909 

.88357 

.48396 

.93785 

.49899 

.99598 

56 

57 

.45463 

.83362 

.46934 

.88445 

.48421 

.93879 

.49924 

.99698 

57 

58 

.45487 

.83444 

.46959 

.88532 

.48446 

.93973 

.49950 

.99799 

58 

59 

.45512 

.83526 

.46983 

.88620 

.48471 

.94066 

.49975 

.99899 

59 

60 

.45536 

.83608 

.47008 

.88708 

.48-496 

.94160  1 

.50000 

.00000 

60 

101 


NATURAL  VERSED  SINES  AND  EXTERNAL  SECANTS, 


6 

0° 

6 

1° 

6 

2" 

6 

3° 

Vers. 

Ex  .se 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.50000 

1.00000 

.51519 

1.06267 

.53053 

1.13005 

.54601 

1.20209 

0 

1 

.50025 

1.00101 

.51544 

1.06375 

.53079 

1.13122 

.54627 

1.20395 

1 

2 

.50050 

1.00202 

.51570 

1.00483 

.53104 

.13239 

.54653 

1.20521 

2 

3 

.50076 

1.00303 

.51595 

1.06592 

.53130 

.13356 

.54679 

1.20647 

3 

4 

.50101 

.00404 

.51621 

1.00701 

.53156 

.13473 

.54705 

1.20773 

4 

5 

.50126 

.00505 

.51646 

.06809 

.53181 

.13590 

.54731 

1.20900 

5 

6 

.50151 

.00607 

.51672 

.06918 

.53207 

.13707 

.54757 

1.21026 

6 

7 

.50176 

.00708 

.51697 

.07027 

.53233 

.13825 

.54782 

1.21153 

7 

8 

.50202 

.00810 

.51723 

.07137 

.53258 

.13942 

.54808 

1.21280 

8 

0 

.50227 

.00912 

.51748 

.07246 

.53284 

.14060 

.54834 

1.21407 

0 

10 

.50252 

.01014 

.51774 

.07356 

.53310 

.14178 

.54860 

1.21535 

10 

11 

.50277 

.01116 

.51799 

.07465 

.53336 

.14296 

.54886 

1.21662 

11 

12 

.50303 

.01218 

.51825 

.07575 

.53301 

.14414 

.54912 

1.21790 

12 

13 

.50328 

.01320 

.51850 

.07685 

.53387 

.14533 

.54938 

1.21918 

13 

14 

.50353 

.01422 

.51876 

.07795 

.53413 

.14651 

.54904 

1.22045 

14 

15 

.50378 

.01525 

.51901 

.07905 

.53439 

.14770 

.54990 

1.22174 

15 

10 

.50404 

.01628 

.51927 

.08015 

.53404 

.14889 

.55016 

1.22302 

10 

17 

.50429 

.01730 

.51952 

.08126 

.53490 

.15008 

.55042 

1.22430 

17 

18 

.50454 

.01833 

.51978 

.08236 

.53516 

.15127 

.55008 

.22559 

18 

19 

.50479 

.01936 

.52003 

.08347 

.53542 

.15246 

.55094 

.22688 

19 

20 

.50505 

.02039 

.52029 

.08458 

.53567 

.15366 

.55120 

.22817 

20 

21 

.50530 

.02143 

.52054 

.08569 

.53593 

.15485 

.55146 

.22946 

21 

22 

.50555 

.02246 

.52080 

.08680 

.53619 

.15605 

.55172 

.23075 

22 

23 

.50581 

.02349 

.52105 

.08791 

.53645 

.15725 

.55198 

.23205 

23 

24 

.50000 

.02453 

.52131 

.08903 

.5367'0 

1.15845 

.55224 

.23334 

21 

25 

.50031 

.02557 

.52156 

.09014 

.53696 

1.15965 

.55250 

.23464 

25 

20 

.50056 

.02661 

.52182 

1.09126 

.53722 

1.16085 

.55276 

.23594 

20 

27 

.50682 

.02765 

.52207 

1.09238 

.53748 

1.10206 

.55302 

.23724 

27 

28 

.50707 

.02869 

.52233 

1.09350 

.53774 

1.16326 

.55328 

.23855 

28 

29 

.50732 

.02973 

.  52259 

1.09462 

.53799 

1.16447 

.55354 

.23985 

29 

30 

.50758 

.03077 

.52284 

1.09574 

.53825 

1.16568 

.55380 

.24116 

30 

31 

.50783 

.03182 

.52310 

1.09686 

.53851 

1.16689 

.55406 

.24247 

31 

32 

.50808 

.03286 

.52335 

1.09799 

.53877 

1.16810 

.55432 

.24378 

32 

33 

.50834 

.03391 

.52361 

1.09911 

.53903 

1.10932 

.55458 

.24509 

33 

34 

.50859 

.03496 

.52386 

1.10024 

.53928 

1.17053 

.55484 

1.24640 

34 

35 

.50884 

.03601 

.52412 

1.10137 

.53954 

1.17175 

.55510 

1.24772 

35 

30 

.50910 

.03706 

.52438 

1.10250 

.53980 

1.17297 

.55536 

1.24903 

36 

37 

.50935 

.03811 

.52463 

1.10363 

.54006 

1.17419 

.55563 

1.25035 

37 

:-;s 

.50960 

.03916 

.52489 

1.10477 

.54032 

1.17541 

.55589 

1.25167 

38 

39 

.50986 

.04022 

.52514 

1.10590 

.54058 

1.17003 

.55015 

1.25300 

39 

40 

.51011 

.04128 

.52540 

1.10704 

.54083 

1.17786 

.55641 

1.25432 

40 

41 

.51036 

.04233 

.52566 

1.10817 

.54109 

1.17909 

.55667 

1.25565 

41 

42 

.51062 

.04339 

.52591 

1.10931 

.54135 

1.18031 

.55093 

1.25697 

42 

43 

.51087 

.04445 

.52617 

1.11045 

.54161 

1.18154 

.55719 

.25830 

43 

44 

.51113 

.04551 

.52642 

1.11159 

.54187 

1.18277 

.55745 

.25963 

44 

45 

.51138 

.04658 

.52668 

1.11274 

.54213 

1.18401 

.55771 

.26097 

45 

40 

.51163 

.04764 

.52694 

1.11388 

.54238 

1.18524 

.55797 

.26230 

40 

47 

.51189 

.04870 

.52719 

1.11503 

.54264 

1.18648 

.55823 

.26364 

47 

48 

.51214 

.04977 

.52745 

1.11617 

.54290 

.18772 

.55849 

.26498 

48 

49 

.51239 

.05084 

.52771 

1.11732 

.54316 

.18895 

.55876 

.26032 

49 

50 

.51265 

.05191 

.52796 

1.11847 

.54342 

.19019 

.55902 

.26766 

50 

51 

.51290 

.05298 

.52822 

1.11963 

.54368 

.19144 

.55928 

.26900 

51 

52 

.51316 

.05405 

.52848 

1.12078 

.54394 

.19268 

.55954 

.27035 

52 

53 

.51341 

.05512 

.52873 

1.12193 

.54420 

.19393 

.55980 

.27169 

53 

54 

.51366 

.05619 

.52899 

1.12309 

.54446 

.19517 

.56006 

.27304 

54 

55 

.51392 

.05727 

.52924 

1.12425 

.54471 

.19642 

.56032 

1.27439 

55 

50 

.51417 

.05835 

.52950 

1.12540 

.54497 

.  19767 

.56058 

1.27574 

50 

57 

.51443 

.05942 

.52976 

1  .  12657 

.54523 

.19892 

.56084 

1.27710 

57 

58 

.51468 

.06050 

.53001 

1.12773 

.54549 

.20018 

.56111 

1.27845 

58 

59 

.51494 

.06158 

.53027 

1.12889 

.54575 

.20143 

.56137 

1.27981 

59 

60 

.51519 

.06267 

.53053 

1.13005 

.54601 

.20269 

.56163 

1.28117 

60 

103 


NATURAL  VERSED  SINES  AND   EXTERNAL  SECANTS, 


G 

40 

6 

5° 

6 

6° 

6 

7° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.56163 

.28117 

.57738 

1.36620 

.59326 

1.45859 

.60927 

1.559J30 

0 

1 

.56189 

.28253 

.57765 

1.36768 

.59353 

1.46020 

.60954 

1.56106 

1 

2 

.56215 

.28390 

.57791 

1.36916 

.59379 

1.46181 

.60980 

1.56282 

2 

3 

.56241 

.28526 

.57817 

1.37064 

.59406 

1.46342 

.61007 

1.56458 

3 

4 

.56267 

.28663 

.57844 

1.37212 

.59433 

1.46504 

.61034 

1.56634 

4 

5 

.56294 

1.28800 

.57870 

1.37361 

.59459 

1.46665 

.61061 

1.56811 

5 

6 

.56320 

1.28937 

.57896 

1.37509 

.59486 

1.46827 

.61088 

1.56988 

6 

7 

.56346 

1.29074 

.57923 

1.37658 

.59512 

1.46989 

.61114 

1.57165 

7 

8 

.56372 

1.29211 

.57919 

1.37808 

.59539 

1.47152 

.61141 

1.57342 

8 

9 

.56398 

1.29349 

.57976 

1.37957 

.59566 

1.47314 

.61168 

1.57520 

9 

10 

.56425 

1.29487 

.58002 

1.38107 

.59592 

1.47477 

.61195 

1.57698 

10 

11 

.56451 

1.29625 

.58028 

1.38256 

.5961D 

1.47640 

.61222 

1.57876 

11 

12 

.56477 

1-29763 

.58055 

1.38406 

.59645 

1.47804 

.61248 

1.58054 

12 

13 

.56503 

1.29901 

.58081 

1.38556 

.59672 

1.47967 

.61275 

1.58233 

13 

11 

.56529 

1.30040 

.58108 

1.38707 

.59699 

1.48131 

.61302 

1.58412 

14 

15 

.56555 

1.30179 

.58134 

1.38857 

.59725 

1.48295 

.61329 

1.58591 

15 

16 

.56582 

1.30318 

.58160 

1.39008 

.59752 

1.48459 

.61356 

1.58771 

1G 

17 

.56608 

1.30457 

.58187 

1.39159 

.59779 

1.48624 

.61383 

1.58950 

17 

18 

.56634 

1.30596 

.58213 

1.39311 

.59805 

1.48789 

.61409 

1.59130 

18 

1!) 

.566(50 

1.30735 

.58240 

1.39462 

.59832 

1.48954 

.61436 

1.59311 

19 

20 

.56687 

1.30875 

.58206 

1.39614 

.59859 

1.49119 

.61403 

1.59491 

20 

m 

.56713 

1.31015 

.58293 

1.39766 

.59885 

1.49284 

.61490 

1.59672 

21 

22 

.56730 

1.31155 

.58319 

1  .33918 

.59912 

1.49450 

.61517 

1.59853 

22 

33 

.56765 

1.31295 

.58345 

1.40070 

.59938 

1.49616 

.61544 

1.60035 

23 

21 

.5679! 

1.31436 

.58372 

1.40222' 

.59965 

1.49782 

.61570 

1.60217 

24 

25 

.56818 

1.31576 

.58398 

1.40375 

.59992 

1.49948 

.61597 

1.60399 

25 

20 

.56844 

1.31717 

..58425 

1.40528 

.60018 

1.50115 

.61624 

1.60581 

2G 

27 

.56870 

1.31858 

.58451 

1.40681 

.60045 

1.50282 

.61651 

1.60763 

27 

28 

.56896 

1.31999 

.58478 

1  .40835 

.60072 

1.50449 

.61678 

1.60946 

28 

2!) 

.56923 

1.32140 

.58504 

1.40988 

.60098 

1.50617 

.61705 

1.61129 

29 

30 

.56949 

1.32282 

.58531 

1.41142 

.60125 

1.50784 

.61732 

1.61313 

30 

31 

.56975 

1.32424 

.58557 

1.41296 

.60152 

1.50952 

.61759 

.61496 

31 

32 

.57001 

1.32566 

.58584 

1.41450 

.60178 

1.51120 

.61785 

.61680 

32 

33 

.57028 

1.32708 

.58610 

1.41605 

.60205 

1.51289 

.61812 

.61864 

33 

34 

.57054 

1.32850 

.58637 

1.41760 

.60232 

1.51457 

.61839 

.62049 

34 

35 

.57080 

1.32993 

.58663 

1.41914 

.60259 

1.51626 

.61866 

.62234 

35 

36 

.57106 

1.33135 

.58690 

1.42070 

.60285 

1.51795 

.61893 

.62419 

36 

37 

.57133 

1.33278 

.58716 

1.42225 

.60312 

1.51965 

.61920 

.62604 

ffl 

38 

.57159 

.33422 

.58743 

1.42380 

.60339 

1.52134 

.61947 

.62790 

38 

39 

.57185 

.33565 

.58769 

1.42536 

.60365 

1.52304 

.61974 

.62976 

39 

40 

.57212 

.33708 

.58796 

1.42692 

.60392 

1.52474 

.62001 

.63162 

40 

41 

.57238 

.33852 

.58822 

1.42848 

.60419 

1.52645 

.62027 

.63348 

41 

42 

.57264 

.33996 

.58849 

1.43005 

.60445 

1.52815 

.62054 

.63535 

42 

43 

.57291 

.34140 

.58875 

1.43162 

.60472 

1.52986 

.62081 

.63722 

43 

44 

.57317 

.34284 

.58902 

1.43318 

.60499 

1.53157 

.62108 

.63909 

44 

45 

.57343 

.34429 

.58928 

1.43476 

.60526 

1.53329 

.62135 

.64097 

45 

46 

.57369 

.34573 

.58955 

1.43633 

.60552 

1.53500 

.62162 

.64285 

46 

47 

.57396 

.34718 

.58981 

1.43790 

.60579 

1.53672 

.62189 

.64473 

47 

48 

.57422 

.34863 

.59008 

1.43948 

.60606 

1.53845 

.62216 

1.64662 

48 

49 

.57448 

.35009 

.59034 

1.44106 

.60633 

1.54017 

.62243 

1.64851 

49 

50 

.57475 

.35154 

.59061 

J.  44264 

.60659 

1.54190 

.62270 

1.65040 

50 

51 

.57501 

.35300 

.59087 

.44423 

.60686 

1.54363 

.62297 

1.65229 

51 

52 

.57527 

.35446 

.59114 

.44582 

.60713 

1.54536 

.62324 

.65419 

52 

53 

.57554 

.35592  ! 

.59140 

.44741 

.00740 

1.54709 

.62351 

.65609 

53 

54 

.57580 

.35738 

.59167 

.44900 

.60766 

1.54883 

.62378 

.65799 

54 

55 

.57606 

.35885 

.59194 

.45059 

.60793 

1.5505? 

.62405 

.65989 

55 

56 

.57633 

.36031 

.59220 

.45219 

.60820 

1.55231 

.62431 

.66180 

56 

57 

.57659 

.36178 

.59247 

.45378 

.60847 

1.55405 

.62458 

.66371 

57 

58 

.57685 

.36325 

.59273 

1.45539 

.60873 

1.55580 

.62485 

.66563 

58 

59 

.57712 

.36473 

.59300 

1.45699 

.60900 

1.55755 

.62512 

.66755 

59 

60 

.57738 

.36620 

.59326 

1.45859 

.60927 

1.55930 

.62539 

.66947 

60 

103 


NATURAL  VERSED  SINES  AND   EXTERNAL  SECANTS. 


/ 

68- 

( 

;o° 

i 

'0° 

i 

n° 

Vers. 

Ex.  sec 

Vers. 

Ex.  sec 

Vers. 

Ex.  sec 

Vers. 

Ex.  sec. 

.62539 

1.66947 

.64163 

1.79043 

.65798 

1.92380 

.67443 

2.07155 

0 

.62566 

1.67139 

.64190 

1.79254 

.65825 

1.92614 

.67471 

2.07415 

] 

.62593 

1.67332 

.64218 

1.79466 

.65853 

1.92849 

.67498 

2.07675 

j 

.62620 

1.67525 

.64245 

1.79679 

.65880 

1.93083 

.67526 

2.07936 

j 

< 

.62647 

1.67718 

.64272 

1.79891 

.65907 

1.93318 

.67553 

2.08197 

i 

.62674 

1.67911 

.64299 

1.80104 

.65935 

1.  935*4 

.67581 

2.08459 

t 

1 

.62701 

1.68105 

.64326 

1.80318 

.65962 

1.93790 

.67608 

2.08721 

6 

' 

.62728 

1.68299 

.64353 

1.80531 

.65989 

1.94026 

.67636 

2.08983 

7 

.62755 

1.68494 

.64381 

1.80746 

.66017 

1.94263 

.67663 

2.09246 

8 

J 

.62782 

1.68689 

.64408 

1.80960 

.66044 

1.94500 

.67691 

2.09510 

9 

10 

.62809 

1.68884 

.64435 

1.81175 

.66071 

1.94737 

.67718 

2.09774 

10 

11 

.62836 

1.69079 

.64462 

1.81390 

.66099 

1.94975 

.67746 

2.10038 

1 

12 

.62863 

1.69275 

.64489 

1.81605 

.66126 

1.95213 

.67773 

2.10303 

<• 

13 

.62890 

1.69471 

.64517 

1.81821 

.66154 

1.95452 

.67801 

2.10568 

j 

14 

.62917 

1.69667 

.64544 

1.82037 

.66181 

1.95691 

.67829 

2.10834 

^ 

15 

.62944 

1.69864 

.64571 

1.82254 

.66208 

1.95931 

.67856 

2.11101 

t 

16 

.62971 

1.70061 

.64598 

1.82471 

.06236 

1.96171 

.67884 

2.11367 

16 

17 

.62998 

1.70258 

.64625 

1.82688 

.66263 

1.96411 

.67911 

2.11635 

17 

18 

.63025 

1.70455 

.64653 

1.8S906 

.66290 

1.96652 

.67939 

2.11903 

18 

19 

.63052 

1.70653 

.64680 

1.83124 

.66318 

1.96893 

.67966 

2.12171 

19 

20 

.63079 

1.70851 

.64707 

1.83342 

.66345 

1.97135 

.67994 

2.12440 

20 

21 

.63106 

1.71050 

.64734 

1.83561 

.66373 

1.97377 

.68021 

2.12709 

21 

22 

.63133 

1.71249 

.64761 

1.83780 

.G6400 

1.97619 

.08049 

2.12979 

22 

2? 

.63161 

1.71448 

.64789 

1.83999 

.66427 

1.97862 

.68077 

2.13249 

23 

24 

.63188 

1.71647 

.64816 

1.84219 

.66455 

1.98106 

.68104 

2.13520 

24 

25 

.63215 

1.71847 

.64843 

1.84439 

.66482 

1.98349 

.68132 

2.13791 

25 

26 

.63243 

1.72047 

.64870 

1.84659 

.66510 

1.98594 

.68159 

2.14063 

26 

27 

.63269 

1.72247 

.04898 

1.84880 

.66537 

1.98838 

.68187 

2.14335 

27 

28 

.63298 

1.7'2448 

.64925 

1.85102 

.66504 

1.99083 

.68214. 

2.14608 

28 

29 

.63323 

1.72649 

.64952 

1.85323 

.66592 

1.99329 

.68242 

2.14881 

29 

30 

.63350 

1.72850 

.64979 

1.85545 

.66619 

1.99574 

.68270 

2.15155 

30 

31 

.63377 

1.73052 

.65007 

1.85767 

.66647 

1.99821 

.68297 

2.15429 

31 

32 

.63404 

1.73254 

.65034 

1.85990 

.66674 

2.00067 

.68325 

2.15704 

32 

33 

.63431 

1.73456 

.65061 

1.86213 

.66702 

2.00315 

.68352 

2.15979 

33 

34 

.63458 

1.73659 

.65088 

1.86437 

.66729 

2.00562 

.68380 

2.16255 

34 

35 

.63485 

1.73862 

.65116 

1.86661 

.66756 

2.00810 

.68408 

2.16531 

35 

36 

.63512 

1.74065 

.65143 

1.86885 

.66784 

2.01059 

.68435 

2.16808 

36 

37 

.63539 

1.74269 

.65170 

1.87109 

.66811 

2.01308 

.68463 

2.17085 

37 

38 

.63566 

1.74473 

.65197 

1.87334 

.66839 

2.01557 

.68490 

2.17363 

38 

39 

.63594 

1.74677 

.65225 

1.87560 

.66866 

2.01807 

.68518 

2.17641 

39 

40 

.63621 

1.74881 

.65252 

1.87785 

.66894 

2.02057 

.68546 

2.17920 

40 

41 

.63648 

1.73086 

.65279 

1.88011 

.66921 

2.02308 

.68573 

2.18199 

41 

42 

.63675 

1.75292 

.65306 

1.88238 

.66949 

2.02559 

.68601 

2.18479 

42 

43 

.63702 

1.75497 

.65334 

1.88465 

.6697'6 

2.02810 

.68628 

2.18759 

43 

44 

.63729 

1.75703 

.65361 

1.88692 

.67003 

2.03062 

.68656 

2.19040 

44 

45 

.63756 

1.75909 

.65388 

1.88920 

.67031 

2.03315 

.68684 

2.19322 

45 

:6 

.63783 

1.76116 

.65416 

1.89148 

.67058 

2.03568 

.68711 

2.19604 

46 

:7 

.63810 

1.76323 

.65443 

1.89376 

.67086 

2.03821 

.68739 

2.19886 

47 

48 

.63838 

1.76530 

.65470 

1.89605 

.67113 

2.04075 

.68767 

2.20169 

48 

49 

.63865 

1.76737 

.65497 

1.89834 

.67141 

2.04329 

.68794 

2.20453 

49 

50 

.63893 

1.76945 

.65525 

1.90063 

.67168 

2.04584 

.68822 

2.20737 

50 

51 

.63919 

1.77154 

.65552 

1.90293 

.67196 

2.04839 

.68849 

2.21021 

51 

52 

.63946 

1.77362 

.65579 

1.90524 

.67223 

2.05094 

.68877 

2.21306 

52 

53 

.63973 

1.77571 

.65607 

1.90754 

.67251 

2.05350 

.68905 

2.21592 

53 

54 

.64000 

1.77780 

.65634 

1.90986 

.67278 

2.05607 

.68932 

2.21878 

54 

55 

.64027 

1.77990 

.65661 

1.91217 

.67306 

2.05864 

.68960 

2.22165 

55 

56 

.64055 

1.78200 

.65689 

1.91449 

.67333 

2.06121 

.68988 

2.22452 

)6 

57 

.64082 

1.78410 

.65716 

1.91681 

.67361 

2.06379 

.69015 

2.22740 

>7 

58 

.64109 

1.78621 

.65743 

1.91914 

.67388 

2.06637 

.69043 

2.23028 

58 

59 

.64136 

1.78832 

.65771 

1.92147 

.67416 

2.06896 

.69071 

2.23317 

59 

60 

.64163 

1.79043 

.65798 

1.92380 

.67443 

2.07155 

.69098 

2.23607 

50 

104 


NATURAL  VERSED  SINES  AND    EXTERNAL  SECANTS. 


> 

72° 

73° 

74° 

75° 

' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.69098 

2.23607 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

0 

1 

.69126 

2.23897 

.70791 

2.42356 

.72464 

2.63164 

.74146 

2.86790 

1 

2 

.69154 

2.24187 

.70818 

2.42683 

.72492 

2.63533 

.74174 

2.87211 

2 

3 

.69181 

2.24478 

.70846 

2.43010 

.72520 

2.63903 

.74202 

2.87633 

3 

4 

.69209 

2.24770 

.70874 

2.43337 

.72548 

2.64274 

.74231 

2.88056 

4 

5 

.69237 

2.25062 

.70902 

2.43666 

.72576 

2.64645 

.74259 

2.88479 

5 

6 

.69264 

2.25355 

.70930 

2.43995 

.72604 

2.65018 

.74287 

2.88904 

6 

7 

.69292 

2.25648 

.70958 

2.44324 

.72632 

2.65391 

.74315 

2.89330 

7 

8 

.69320 

2.25942 

.70985 

2.44655 

.72660 

2.65765 

.74343 

2.89756 

8 

9 

.69347 

2.26237 

.71013 

2.44986 

.72688 

2.66140 

.74371 

2.90184 

9 

10 

.69375 

2.26531 

.71041 

2.45317 

.72716 

2.66515 

.74399 

2.90613 

10 

11 

.69403 

2.26827 

.71069 

2.45650 

.72744 

2.66892 

.74427 

2.91042 

11 

13 

.69430 

2.27123 

.71097 

2.45983 

.72773 

2.67269 

.74455 

2.91473 

12 

13 

.69458 

2.27420 

.71125 

2.46316 

.72800 

2.67647 

.74484 

2.91904 

13 

14 

.69486 

2.27717 

.71153 

2.46651 

.72828 

2.68025 

.74512 

2.92337 

14 

15 

.69514 

2.28015 

.71180 

2.46986 

.72856 

2.68405 

.74540 

2.92770 

15 

16 

.69541 

2.28313 

.71208 

2.47321 

.72884 

2.68785 

.74568 

2.93204 

16 

17 

.69569 

2.28612 

.71236 

2.47658 

.72912 

2.69167 

.74596 

2.93640 

17 

18 

.69597 

2.28912 

.71264 

2.47995 

.72940 

2.69549 

.74624 

2.94076 

18 

19 

.69624 

2.29212 

.71292 

2.48333 

.72968 

2.69931 

.74652 

2.94514 

19 

20 

.69652 

2.29512 

.71320 

2.48671 

.72996 

2.70315 

.74680 

2.94952 

20 

31 

.69680 

2.29814 

.71348 

2.49010 

.73024 

2.70700 

.74709 

2  95392 

21 

22 

.69708 

2.30115 

.71375 

2.49350 

.73052 

2.71085 

.74737 

2^95832 

22 

28 

.69735 

2.30418 

.71403 

2.49691 

.73080 

2.71471 

.74765 

2.96274 

23 

24 

.69763 

2.30721 

.71431 

2.50032 

.73108 

2.71858 

.74793 

2.96716 

24 

25 

.69791 

2.31024 

.71459 

2.50374 

.73136 

2.72246 

.74821 

2.97160 

25 

20 

.69818 

2.31328 

.71487 

2.50716 

.73164 

2.72635 

.74849 

2.97604 

20 

27 

.69846 

2.31633 

.71515 

2.51060 

.73192 

2.73024 

.74878 

2.98050 

27 

28 

.69874 

2.31939 

.71543 

2.51404 

.73220 

2.73414 

.74906 

2.98497 

28 

29 

.69902 

2,32244 

.71571 

2.51748 

.73248 

2.73806 

.74934 

2.98944 

29 

30 

.69929 

2.32551 

.71598 

2.52094 

.73276 

2.74198 

.74962 

2.99393 

30 

31 

.69957 

2.32858 

.71626 

2.52440 

.73304 

2.74591 

.74990 

2.99843 

31 

32 

.69985 

2.33166 

.71654 

2.52787 

.73332 

2.74984 

.75018 

3.00293 

32 

33 

.70013 

2.33474 

.71682 

2.53134 

.73360 

2.75379 

.75047 

3.00745 

33 

34 

.70040 

2.33783 

.71710 

2.53482 

.73388 

2.75775 

.75075 

3.01198 

34 

35 

.70068 

2.34092 

.71738 

2.53831 

.73416 

2.76171 

.75103 

3.01652 

35 

36 

.70096 

2.34403 

.  71766 

2.54181 

.73444 

2.76568 

.75131 

3.02107 

36 

37 

.70124 

2.34713 

.71794 

2.54531 

.73472 

2.76966 

.75159 

3.02563 

37 

38 

.70151 

2.35025 

.71822 

2.54883 

.73500 

2.77365 

.75187 

3.03020 

38 

39 

.70179 

2.35336 

.71850 

2.55235 

.73529 

2.77765 

.75216 

3.03479 

39 

40 

.70207 

2.35649 

.71877 

2.55587 

.73557 

2.78166 

.75244 

3.03938 

40 

41 

.70235 

2.35962 

.71905 

2.55940 

.73585 

2.78568 

.75272 

3.04398 

41 

42 

.70263 

2.36276 

.71933 

2.56294 

.73613 

2.78970 

.75300 

3.04860 

42 

43 

.70290 

2.3G590 

.71961 

2.56649 

.7'3641 

2.79374 

.75328 

3.05322 

43 

44 

.70318 

2.36905 

.71989 

2.57005 

.73669 

2.7977S 

.75356 

3.05786 

44 

45 

.70346 

2.37221 

.72017 

2.57361 

.73697 

2.80183 

.75385 

3.06251 

45 

46 

.70374 

2.37537 

.72045 

2.57718 

.73725 

2.80589 

.75413 

3.06717 

40 

47 

.70401 

2.37854 

.72073 

2.58076 

.73753 

2.80996 

.75441 

3.07184 

47 

48 

.70429 

2.38171 

.72101 

2.58434 

.73781 

2.81404 

.75469 

3.07652 

48 

49 

.70457 

2.38489 

.72129 

2.58794 

.73809 

2.81813 

.75497 

3.08121 

49 

50 

.70485 

2.38808 

.72157 

2.59154 

.73837 

2.82223 

.75526 

3.08591 

50 

51 

.70513 

2.39128 

.72185 

2.59514 

.73865 

2.82633 

.75554 

3.09063 

51 

52 

.70540 

2.39448 

.72213 

2.59876 

.73893 

2.83045 

.75582 

3.09535 

52 

53 

.70568 

2.39768 

.72241 

2.60238 

.73921 

2.83457 

.75610 

3.10009 

53 

54 

.70596 

2.40089 

.72269 

2.60601 

.73950 

2.83871 

.75639 

3.10484 

54 

55 

.70624 

2.40411 

.72296 

2.60965 

.73978 

2.84285 

.75667 

3.10960 

55 

56 

.70652 

2.40734 

.72324 

2.61330 

.74006 

2.84700 

.75695 

3.11437 

50 

57 

.70679 

2.41057 

.72352 

2.61695 

.74034 

2.85116 

.75723 

3.11915 

57 

58 

.70707 

2.41381 

.72380 

2.62061 

.74062 

2.85533 

.75751 

3.12394 

58 

59 

.70735 

2.41705 

.72408 

2.62428 

.74090 

2.85951 

.75780 

3.12875 

59 

GO 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

.75808 

3.13357 

60 

105 


NATURAL  VERSED  SINES  AND  EXTERNAL    SECANTS. 


' 

76° 

77° 

78° 

79° 

' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.75808 

3.13357 

.77505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

0 

1 

.75836 

3.13839 

.77533 

3.45102 

.79237 

3.81633 

.80948 

4.24870 

1 

2 

.75864 

3.14323 

.77562 

3.45664 

.79266 

3.82294 

.80976 

4.25658 

2 

3 

.75892 

3.14809 

.77590 

3.46228 

.79294 

3.82956 

.81005 

4.26448 

3 

4 

.75921 

3.15295 

.77618 

3.46793 

.79323 

3.83621 

.81033 

4.27241 

4 

5 

.75949 

3.15782 

.77647 

3.47360 

.79351 

3.84288 

.81062 

4.28036 

5 

6 

.75977 

3.16271 

.77675 

3.47928 

.79380 

3.84956 

.81090 

4.28833 

6 

7 

.76005 

3.16761 

.77703 

3.48498 

.79408 

3.85627 

.81119 

4.29634 

7 

8 

.76034 

3.17252 

.77732 

3.49069 

.79437 

3.86299 

.81148 

4.30436 

8 

9 

.76062 

3.17744 

.77760 

3.49642 

.79465 

3.86973 

.81176 

4.31241 

9 

10 

.76090 

3.18238 

.77788 

3.50216 

.79493 

3.87649 

.81205 

4.32049 

10 

11 

.76118 

3.18733 

.77817 

3.50791 

.79522 

3.88327 

.81233 

4.32859 

11 

12 

.76147 

3.19228 

.77845 

3.51368 

.79550 

3.89007 

.81262 

4  .  33671 

12 

13 

.76175 

3.19725 

.77874 

3.51947 

.79579 

3.89689 

.81290 

4.34486 

13 

14 

.76203 

3.20224 

.77902 

3.52527 

.79607 

3.90373 

.81319 

4.35304 

14 

15 

.76231 

3.20723 

.77930 

3.53109 

.79636 

3.91058 

.81348 

4.36124 

15 

16 

.76260 

3.21224 

.77959 

3.53692 

.79664 

3.91746 

.81376 

4.36947 

10 

17 

.76288 

3.21726 

.77987 

3.54277 

.79693 

3.92436 

.81405 

4.37772 

IT 

18 

.76316 

3.22229 

.78015 

3.54863 

.79721 

3.93128 

.81433 

4.38600 

18 

19 

.76344 

3.22734 

.78044 

3.55451 

.79750 

3.93821 

.81402 

4.39430 

11) 

20 

.76373 

3.23239 

.78072 

3.56041 

.79778 

3.94517 

.81491 

4.40263 

20 

21 

.76401 

3.23746 

.78101 

3.56632 

.79807 

3.95215 

.81519 

4.41099 

21 

22 

.76429 

3.24255 

.78129 

3.57224 

.79835 

3.95914 

.81548 

4.41937 

22 

23 

.76458 

3.24764 

.78157 

3.57819 

.79864 

3.96616 

.81576 

4.42778 

23 

24 

.76486 

3.25275 

.78186 

3.58414 

.79892 

3.97320 

.81605 

4.43622 

24 

25 

.76514 

3.25787 

.78214 

3.59012 

.79921 

3.98025 

.81633 

4.44468 

25 

20 

.76542 

3.26300 

.78242 

3.59611 

.79949 

3.98733 

.81662 

4.45317 

26 

27 

.76571 

3.26814 

.78271 

3.60211 

.79978 

3.99443 

.81691 

4.46169 

27 

28 

.76599 

3.27330 

.78299 

3.60813 

.80006 

4.00155 

.81719 

4.47023 

28 

29 

.76627 

3.27847 

.78328 

3.61417 

.80035 

4.00869 

.81748 

4.47881 

29 

30 

.76655 

3.28366 

.  7835S 

3.62023 

.80063 

4.01585 

.81776 

4.48740 

30 

a 

.76684 

3.28885 

.78384 

3.62630 

.80092 

4.02303 

.81805 

4.49603 

31 

32 

.76712 

3.29406 

.78413 

3.63238 

.80120 

4.03024 

.81834 

4.50468 

32 

33 

.76740 

3.29929 

.78441 

3.63849 

.80149 

4.03746 

.81862 

4.51337 

33 

34 

.76769 

3.30452 

.78470 

3.64461 

.80177 

4.04471 

.81891 

4.52208 

34 

35 

.76797 

3.30977 

.78498 

3.65074 

.80205 

4.05197 

.81919 

4.53081 

35 

30 

.76825 

3.31503 

.78526 

3.65690 

.80234 

4.05926 

.81948 

4.53958 

36 

37 

.76854 

3.32031 

.7'8555 

3.66307 

.80263 

4.06657 

.81977 

4.54837 

37 

38 

.76882 

3.32500 

.78583 

3.66925 

.80291 

4.07390 

.82005 

4.55720 

38 

39 

.76910 

3.33090 

.78612 

3.67545 

.80320 

4.08125 

.82034 

4.56605 

39 

40 

.76938 

3.33622 

.78640 

3.68167 

.80348 

4.08863 

.82063 

4.57493 

40 

41 

.76967 

3.34154 

.78669 

3.68791 

.80377 

4.09602 

.82091 

4.58383 

41 

42 

.76995 

3.34689 

.78697 

3.69417 

.80405 

4.10344 

.82120 

4.59277 

42 

43 

.77023 

3.35224 

.78725- 

3.70044 

.80434 

4.11088 

.82148 

4.60174 

43 

44 

.77052 

3.35761 

.78754 

3.70673 

.80462 

4.11835 

.82177 

4.61073 

44 

45 

.77080 

3.36299 

.78782 

3.71303 

.80491 

4.12583 

.82206 

4.61976 

45 

40 

.77108 

3.36839 

.78811 

3.71935 

.80520 

4.13334 

.82234 

4.62881 

46 

47 

.77137 

3.37380 

.78839 

3.72569 

.80548 

4.14087 

.82263 

4.63790 

47 

48 

.77165 

3.37923 

.78868 

3.73205 

.80577 

4.14842 

.82292 

4.64701 

48 

49 

.77193 

3.38466 

.78896 

3.73843 

.80605 

4.15599 

.82320 

4.65616 

49 

50 

.77222 

3.39012 

.78924 

3.74482 

.80634 

4.16359 

.82349 

4.66533 

50 

51 

.77250 

3.39558 

.78953 

3.75123 

.80662 

4.17121 

.82377 

4.67454 

51 

52 

.77278 

3.40106 

.78981 

3.75760 

.80691 

4.17886 

.82406 

4.68377 

52 

53 

.77307 

3.40656 

.79010 

3.76411 

.80719 

4.18652 

.82435 

4.69304 

53 

54 

.77335 

3.41206 

.79038 

3.77057 

.80748 

4.19421 

.82403 

4.70234 

54 

55 

.77363 

3.41759 

.79067 

3.77705 

.80776 

4.20193 

824?)'> 

4.71166 

55 

56 

.77392 

3.42312 

.79095 

3.78355 

.80805 

4.20966 

'.82521 

4.72102 

56 

57 

.77420 

3.42867 

.79123 

3.79007 

.80833 

4.21742 

.82549 

4.73041 

57 

58 

.77448 

3.43424 

.79152 

3.79661 

.80862 

4.22521 

.82578 

4.73983 

58 

59 

.77477 

3.43982 

.79180 

3.80316 

.80891 

4.23301 

.82607 

4.74929 

59 

60 

.77505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

.82635 

4.75877 

60 

106 


NATURAL  VERSED    SINES   AND  EXTERNAL  SECANTS. 


80° 

81° 

82° 

83° 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.82635 

4.75877 

.84357 

5.39245 

.86083 

6.18530 

.87813 

7.20551 

0 

1 

.82664 

4.76829 

.84385 

5.40422 

.86112 

6.20020 

.87842 

7.22500 

1 

2 

.82692 

4.77784 

.84414 

5.41602 

.86140 

6.21517 

.87871 

7.24457 

2 

3 

.82721 

4.78742 

.84443 

5.42787 

.86109 

6.23019 

.87900 

7.26425 

3 

4 

.82750 

4.79703 

.84471 

5.43977 

.86198 

6.24529 

.87929 

7.28402 

4 

5 

.82778 

4.80667 

.84500 

5.45171 

.86227 

6.20044 

.87957 

7.30388 

5 

6 

.82807 

4.81G35 

.84529 

5.46369 

.86256 

6.27566 

.87986 

7.32384 

6 

7 

.82836 

4.82606 

.84558 

5.47572 

.86284 

6.29095 

.88015 

7.34390 

7 

8 

.82864 

4.83581 

.84586 

5.48779 

.86313 

6.30630 

.88044 

7.36405 

8 

9 

.82893 

4.84558 

.84615 

5.49991 

.86342 

6.32171 

.88073 

7.38431 

9 

10 

.82922 

4.85539 

.84644 

5.51208 

.86371 

6.33719 

.88102 

7.40466 

10 

11 

.82950 

4.86524 

.84673 

5.52429 

.86400 

6.35274 

.88131 

7.42511 

11 

12 

.82979 

4.87511 

.84701 

5.53G55 

.80428 

6.36835 

.88160 

7.44566 

12 

18 

.83008 

4.88502 

.84730 

5.54886 

.86457 

6.38403 

.88188 

7.46632 

13 

14 

.8:3036 

4.89497 

.84759 

5.56121 

.86486 

6.39978 

.88217 

7.48707 

14 

15 

.83065 

4.90495 

.84788 

5.57361 

.86515 

6.41560 

.88246 

7.50793 

15 

1G 

.8:5094 

4.91496 

.84816 

5.58606 

.86544 

6.43148 

.88275 

7.52889 

16 

17 

.83122 

4.92501 

.84845 

5.59855 

.86573 

6.44743 

.88304 

7.54996 

17 

18 

.83151 

4.93509 

.84874 

5.61110 

.86601 

6.46346 

.88333 

7.57113 

18 

1!) 

.83180 

4.94521 

.84903 

5.62369 

.86630 

6.47955 

.88362 

7.59241 

1!) 

20 

.83208 

4.95536 

.84931 

5.63G33 

.86659 

C.49571 

.88391 

7.61379 

20 

2! 

.83237 

4.96555 

.84960 

5.64902 

.86688 

6.51194 

.88420 

7.63528 

21 

oo 

.83366 

4.97577 

.84989 

5.6G1?'6 

.86717 

6.52825 

.88448 

7.65688 

22 

£3 

.83294 

4.98603 

.85018 

5.67454 

.86746 

6.54462 

.88477 

7.67859 

23 

24 

.83323 

4.99633 

.85046 

5.68738 

.86774 

6.56107 

.88506 

7.70041 

21 

25 

.8:3352 

5.00666 

.85075 

5.70027 

.86803 

6.57759 

.88535 

7.72234 

25 

20 

.83380 

5.01703 

.85104 

5.71321 

.86832 

6.59418 

.88564 

7.74438 

20 

27 

.83409 

5.02743 

.85133 

5.72620 

.86861 

6.61085 

.88593 

7.76653 

27 

28 

.8:3438 

5.03787 

.85162 

5.73924 

.86890 

6.62759 

.88622 

7.78880 

28 

29 

.83467 

5.04834 

.85190 

5.75233 

.86919 

6.64441 

.88651 

7.81118 

2!) 

30 

.83405 

5.05886 

.85219 

5.76547 

.8694T 

G.66130 

.88680 

7.83367 

30 

31 

.83524 

5.06941 

.85248 

5.77866 

.86976 

6.67826 

.88709 

7.85628 

31 

32  '|  .83553 

5.08000 

.85277 

5.79191 

.87005 

6.09530 

.88737 

7.87901 

32 

33 

.83581 

5.09062 

.85305 

5.80521 

.87034 

6.71242 

.88766 

7.90186 

33 

34 

.83610 

5.10129 

.85334 

5.81856 

.87063 

6.72962 

.88795 

7.92482 

34 

35 

.83639 

5.11199 

.85363 

5.83196 

.87092 

6.74689 

.88824 

7.94791 

35 

36 

.83667 

5.12273 

.85392 

5.84542 

.87120 

6.76424 

.88853 

7.87111 

30 

37 

.83696 

5.13350 

.85420 

5.85893 

.87149 

6.78167 

.88882 

7.99444 

37 

38 

.83725 

5.14432 

.85449 

5.87250 

.87178 

6.79918 

.88911 

8.01788 

88 

39 

.83754 

5.15517 

.85418 

5.88612 

.87207 

6.81677 

.88940 

8.04146 

30 

40 

.83782 

5.16607 

.85507 

5.89979 

.87236 

6.83443 

.88969 

8.06515 

40 

41 

.83811 

5.17700 

.85536 

5.91352 

.87265 

6.85218 

.88998 

8.08897 

41 

42 

.83840 

5.18797 

.85564 

5.92731 

.87294 

6.87001 

.89027 

8.11292 

42 

43 

.83868 

5.19898 

.85593 

5.94115 

.87322 

6.88792 

.89055 

8.13699 

43 

44 

.83897 

5.21004 

.85622 

5.95505 

.87351 

6.90592 

.89084 

8.16120 

44 

45 

.83926 

5.22113 

.85651 

5.96900 

.87380 

6.92400 

.89113 

8.18553 

45 

46 

.83954 

5.23226 

.85680 

5.9&301 

.87409 

6.94216 

.89142 

8.20999 

40 

47 

.83983 

5.24343 

.85708 

5.99708 

.87438 

6.96040 

.89171 

8.23459 

47 

48 

.84012 

5.25464 

.85737 

6.01120 

.87467 

6.97873 

.89200 

8.25931 

48 

49 

.84041 

5.26590 

.85766 

6.02538 

.87496 

6.99714 

.89229 

8.28417 

49 

50 

.84069 

5.27719 

.85795 

6.03962 

.87524 

7.01565 

.89258 

8.30917 

50 

51 

.84098 

5.28853 

.&5823 

6.05392 

.87553 

7.03423 

.89287 

8.33430 

51 

52 

.84127 

5.29991 

.85852 

6.06828 

.87582 

7.05291 

.89316 

8.35957 

52 

53 

.84155 

5.31133 

.85881 

6.08269 

.87611 

7.07167 

.89345 

8.38497 

53 

54 

.84184 

5.32279 

.85910 

6.09717 

.87640 

7.09052 

.89374 

8.41052 

54 

55 

.84213 

5.33429 

.85939 

6.11171 

.87669 

7.10946 

.89403 

8.43620 

55 

56 

.84242 

5.34584 

.85967 

6.12630 

.87698 

7.12849 

.89431 

8.46203 

50 

57 

.84270 

5.35743 

.85996 

6.14096 

.87726 

7.14760 

.89460 

8.48800 

57 

58 

.84299 

5.36906 

.86025 

6.15568 

.87755 

7.16681 

.89489 

8.51411 

58 

50 

.84328 

5.38073 

.86054 

6.17046 

.87784 

7.18612 

.89518 

8.54037 

59 

60 

.84357 

5.39245 

.86083 

6.18530 

.87813 

7.20551 

.89547 

8.56677 

60 

107 


NATURAL  VERSED  SINES  AND  EXTERNAL   SECANTS. 


/ 

84° 

85° 

86° 

' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.89547 

8.56677 

.91284 

10.47371 

.93024 

13.33559 

0 

1 

.89576 

8.59332 

.91313 

10.51199 

.93053 

13.39547 

1 

2 

.89605 

8.62002 

.91342 

10.55052 

.93082 

13.45586 

2 

3 

.89634 

8.64687 

.91371 

10.58932 

.93111 

13.51676 

3 

4 

.89663 

8.67387 

.91400 

10.62837 

.93140 

13.57817 

4 

5 

.89692 

8.70103 

.91429 

10.66769 

.93169 

13.64011 

5 

6 

.89721 

8.72833 

.91458 

10.70728 

.93198 

13.70258 

6 

7 

.89750 

8.75579 

.91487 

10.74714 

.93227 

13.76558 

7 

8 

.89779 

8.78341 

.91516 

10.78727 

.93257 

13.82913 

8 

9 

.89808 

8.81119 

.91545 

10.82768 

.93286 

13.89323 

9 

10 

.89836 

8.83912 

.91574 

10.86837 

.93315 

13.95788 

10 

11 

.89865 

8.86722 

.91603 

10.90934 

.93344 

14.02310 

11 

12 

.89894 

8.89547 

.91632 

10.95060 

.93373 

14.08890 

12 

13 

.89923 

8.92389 

.91661 

10.99214 

.93402 

14.15527 

13 

14 

.89952 

8.95248 

.91690 

11.03397 

.93431 

14.22223 

14 

15 

.89981 

8.98123 

.91719 

11.07610 

.93460 

14.28979 

15 

16 

.90010 

9.01015 

.91748 

11.11852 

.93489 

14.35795 

16 

17 

.90039 

9.03923 

.91777 

11.16125 

.93518 

14.42672 

17 

18 

.90068 

9.06849 

.91806 

11.20427 

.93547 

14.49611 

18 

19 

.90097 

9.09792 

.91885 

11.24761 

.93576 

14.56614 

19 

20 

.90126 

9.12752 

.91804 

11.29125 

.93605 

14.63679 

20 

21 

.90155 

9.15730 

.91893 

11.33521 

.93634 

14.70810 

21 

22 

.90184 

9.18725 

.91922 

11.37948 

.93663 

14.78005 

22 

23 

.90213 

9.21739 

.91951 

11.42408 

.93692 

14.85268 

23 

24 

.90242 

9.24770 

.91980 

11.46900 

.93721 

14.92597 

24 

25 

.90271 

9.27819 

.92009 

11.51424 

.93750 

14.99995 

25 

26 

.90300 

9.30887 

.92038 

11.55982 

.93779 

15.07462 

26 

27 

.90329 

9.33973 

.92067 

11.60572 

.93808 

15.14999 

27 

28 

.90358 

9.37077 

.92096 

11.65197 

.93837 

15.22607 

28 

29 

.90386 

9.40201 

.92125 

11.69856 

.93866 

15.30287 

29 

30 

.90415 

9.43343 

.92154 

11.74550 

.93895 

15.38041 

30 

31 

.90444 

9.46505 

.92183 

11.79278 

.93924 

15.45869 

31 

32 

.90473 

9.49685 

,92212 

11.84042 

.93953 

15.53772 

32 

33 

.90502 

9.52886 

.92241 

11.88841 

.93982 

15.61751 

33 

34 

.90531 

9.56106 

.92270 

11.93677 

.94011 

15.69808 

34 

35 

.90560 

9.59346 

.92299 

11.98549 

.94040 

15.77944 

35 

36 

.90589 

9.62605 

.92328 

12.03458 

.94069 

15.86159 

36 

37 

.90618 

9.65885 

.92357 

12.08040 

.94098 

15.94456 

37 

38 

.90647 

9.69186 

.92386 

12.13388 

.94127 

16.02835 

38 

39 

.90676 

9.72507 

.92-115 

12.18411 

.94156 

16.11297 

39 

40 

.90705 

9.75849 

.92444 

12.23472 

.94186 

16.19843 

40 

41 

.90734 

9.79212 

.92473 

12.28572 

.94215 

16.28476 

41 

42 

.90763 

9.82596 

.92502 

12.33712 

.94244 

16.37196 

42 

43 

.90792 

9.86001 

.92531 

12.38891 

.94273 

16.46005 

43 

44 

.90821 

9.89428 

.92500 

12.44112 

.94302 

16.54903 

44 

45 

.90850 

9.92877 

.92589 

12.49373 

.94331 

16.63893 

45 

46 

.90879 

9.96348 

.92618 

12.54676 

.94360 

16.72975 

46 

47 

.90908 

9.99841 

.92647 

12.60021 

.94389 

16.82152 

47 

48 

.90937 

10.03356 

.92676 

12.65408 

.94418 

16.91424 

48 

49 

.90966 

10.06894 

.92705 

12.7'0838 

.94447 

17.00794 

49 

50 

.90995 

10.10455 

.92734 

12.76312 

.94476 

17.10262 

50 

51 

.91024 

10.14039 

.92763 

12.81829 

.94505 

17.19830 

51 

52 

.91053 

10.17646 

.92792 

12.87391 

.94534 

17.29501 

52 

53 

.91082 

10.21277 

.92821 

12.92999 

.94563 

17.39274 

53 

54 

.91111 

10.24932 

.92850 

12.98651 

.94592 

17.49153 

54 

55 

.91140 

10.28610 

.92879 

13.04350 

.94621 

17.59139 

55 

56 

.91169 

10.32313 

.92908 

13.10096 

.94650 

17.69233 

56 

57 

.91197 

10.36040 

.92937 

13.15889 

.94679 

17.79438 

57 

58 

.91226 

10.39792 

.92966 

13.21730 

.94708 

17.89755 

58 

59 

.91255 

10.43569 

.92995 

13.27620 

.94737 

18.00185 

59 

60 

.91284 

10.47371 

.93024 

13.33559 

.94766 

18.10732 

60 

108 


NATURAL  VERSED  SINES  AND  EXTERNAL  SEOANT8. 


/ 

87° 

88° 

89° 

/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

0 

.94766 

18-J0732 

.96510 

27.65371 

.98255 

56.29869 

0 

1 

.94795 

18.21397 

.96539 

27.89440 

.98284 

57.26976 

1 

2 

.94825 

18-32182 

.96568 

28.13917 

.98313 

58.27431 

2 

3 

.94854 

18.43088 

.96597 

28.38812 

.98342 

59.31411 

3 

4 

.94883 

18.54119 

.96626 

28.64137 

.98371 

60.39105 

4 

5 

.94912 

18.65275 

.96655 

28.89903 

.98400 

61.50715 

5 

6 

.94941 

18.76560 

.96684 

29.16120 

.98429 

62.66460 

6 

7 

.94970 

18.87976 

.96714 

29.42802 

.98458 

63.86572 

7 

8 

.94999 

18.99524 

.96743 

29.69960 

.98487 

65.11304 

8 

9 

.95028 

19.11208 

.96772 

29.97607 

.98517 

66.40927 

9 

10 

.95057 

19.23028 

.96801 

30.25758 

.98546 

67.75736 

10 

11 

.95086 

19.34989 

.96830 

30.54425 

.98575 

69.16047 

11 

12 

.95115 

19.47093 

.96859 

30.83623 

.98604 

70.62285 

12 

13 

.95144 

19.59341 

.96888 

31.13366 

.98633 

72.14583 

13 

14 

.95173 

19.71737 

.96917 

31.43671 

.98662 

73.73586 

14 

15 

.95202 

19.84283 

.96946 

31.74554 

.98691 

75.39655 

15 

16 

.95231 

19.96982 

.96975 

32.06030 

.98720 

77.13274 

16 

17 

.95260 

20.09838 

.97004 

32.38118 

.98749 

78.94968 

17 

18 

.95289 

20.22852 

.97033 

32.70835 

.98778 

80.85315 

18 

19 

.95318 

20.36027 

.97062 

33.04199 

.98807 

82.84947 

19 

20 

.95347 

20.49368 

.97092 

33.38232 

.98836 

84.94561 

20 

21 

.95377 

20.62876 

.97121 

33.72952 

.98866 

87.14924 

21 

22 

.95406 

20.76555 

.97150 

84.08380 

.98895 

89.46886 

22 

23 

.95435 

20.90403 

.97179 

34.44530 

.98924 

91.91387 

23 

24 

.95464 

21.04440 

.97208 

34.81452 

.98953 

94.49471 

24 

25 

.95493 

21.18653 

.97237 

35.19141 

.98982 

97.22303 

25 

26 

.95522 

.21.33050 

.97266 

35.57633 

.99011 

100.1119 

26 

27 

.95551 

21.47635 

.97295 

35.96953 

.99040 

103.1757 

27 

28 

.95580 

21.62413 

.97324 

36.37127 

.99069 

106.4311 

28 

29 

.95609 

21.77386 

.97353 

36.78185 

.99098 

109.8966 

29 

30 

.95638 

21.92559 

.97382 

37.20155 

.99127 

113.5930 

30 

31 

.95667 

22.07935 

.97411 

37.63068 

.C9156 

117.5444 

31 

32 

.95696 

22.23520 

.97440 

38.06957 

.99186 

121.7780 

32 

33 

.95725 

22.39316 

.97470 

38.51855 

.99215 

126.3253 

33 

34 

.95754 

22.55329 

.97499 

38.97797 

.99244 

131.2223 

34 

35 

.95783 

22.71563 

.97528 

39.44820 

.99273 

136.5111 

35 

36 

.95812 

22.88022 

.97557 

39.92963 

.99302 

142.2406 

36 

37 

.95842 

23.04712 

.97586 

40.42266 

.99331 

148.4684 

37 

38 

.95871 

23.21637 

.97615 

40.92772 

.99360 

155.2623 

38 

39 

.95900 

23.38802 

.97644 

41.44525 

.99389 

162.7033 

39 

40 

.95929 

23.56212 

.97673 

41.97571 

.99418 

170.8883 

40 

41 

.95958 

23.73873 

.97702 

42.51961* 

.99447 

179.9350 

41 

42 

.95987 

23.91790 

.97731 

43.07746 

.99476 

189.9868 

42 

43 

.96016 

24.09969 

.97760 

43.64980 

.99505 

201.2212 

43 

44 

.96045 

24.28414 

.97789 

44.23720 

.99535 

213.8600 

44 

45 

.96074 

24.47134 

.97819 

44.84026 

.99564 

228.1839 

45 

46 

.96103 

24.66132 

.97848 

45.45963 

.99593 

244.5540 

46 

47 

.96132 

24.85417 

.97877 

46.09596 

.99622 

263.4427 

47 

48 

.96161 

25.04994 

.97906 

46.74997 

.99651 

285.4795 

48 

49 

.96190 

25.24869 

.97935 

47.42241 

.99680 

311.5230 

49 

50 

.96219 

25.45051 

.97964 

48.11406 

.99709 

342.7752 

50 

51 

.96248 

25.65546 

.97993 

48.82576 

.99738 

380.9723 

51 

52 

.96277 

25.86360 

.98022 

49.55840 

.99767 

428.7187 

52 

53 

.96307 

26.07503 

.98051 

50.31290 

.99796 

490.1070 

53 

54 

.96336 

26.28981 

.98080 

51.09027 

.89825 

671.9581 

54 

55 

.96365 

26.50804 

.98109 

51.89156 

.99855 

686.5496 

55 

56 

.96394 

26.72978 

.98138 

52.71790 

.99884 

858.4369 

56 

57 

.96423 

26.95513 

.98168 

53.57046 

.99913 

1144.916 

57 

58 

.96452 

27.18417 

.98197 

54.45053 

.99942 

1717.874 

58 

59 

.96481 

27.41700 

.98226 

55.35946 

.99971 

3436.747 

59 

60 

.96510 

27.65371 

.98255 

56.29869 

1.00000 

Infinite 

60 

109 


